 Let's get started with the session. So today I'm going to talk about how do we find the domain and range for certain real valued functions, right? So we are going to talk about domain and range, range, range for the real valued functions that you're going to come across. This entire exercise will take us to classes, because there are so many real valued functions that we need to address. So what are the real valued functions whose domain and range we will be focusing on? And yes, while doing so, we'll be also learning certain inequalities, because inequalities will help us to find the domain and range in some of the cases. So I just list down what are the real valued functions? By the way, the word real valued, I may not be using every time. I'm just using it to begin with. But as the course progresses, I will just use the word functions. I will not use the word real valued functions. By default, all the functions that we are going to deal with, they will be real valued in nature. Okay, real valued function means what can anybody tell me that functions whose both domain and range will be subsets of real numbers. So we will be not talking about any such function where the function takes in or the domain of the function has some non real quantities. Neither will be talking about any such function whose output or whose members of the range will be non real in nature. So what are the functions that we are going to address under real valued functions? So we are going to talk about majorly three types of functions. Okay, so this is for your entire class 11th and 12th. So most of your functions will lie in these categories only. So what are they? So first we'll talk about the algebraic functions. Okay, so what are we going to cover under algebraic functions? Under algebraic functions, we are going to cover up polynomials. Okay, we are going to cover up rational functions. By the way, polynomials are types of rational functions. We will discuss about it in some time. And then we'll talk about irrational functions. So three types of functions we are going to cover up. You can say primarily two types only because polynomials are anyways covered in the rational. So we are going to cover rational and irrational functions under algebraic functions. Next we are going to take up transcendental functions. Transcendental functions. So what are transcendental functions? Non-algebraic functions are called transcendental functions. So we are also going to touch upon domain and range for certain transcendental functions. What are they? Exponential functions. You must be aware of this function because if you have attended the bridge course, we had briefly talked about exponential functions also. We'll be talking about logarithmic functions. Logarithmic functions. And even though I'm listing it over here, we will not talk about trigonometric functions. Now, why we will not talk about trigonometric functions? Any guesses? Why we are not going to talk about domain and range of trigonometric functions? Because anyways, this is going to be covered under trigonometry part. So this part, I'm going to cover under trigonometry. So I don't want the same topics to be repeated over and over again. So this will be covered under trigonometry. So immediately after functions, we'll start with trigonometry. And then we are going to talk about some special functions. We are going to talk about some special functions. These functions you may not have come across in your study of mathematics so far. So in ninth and 10th, we have not discussed this function in our CBC curriculum ever. So what are these new special functions that we are going to learn? So we are going to learn about modulus function. Which includes a bit of idea about modulus inequality also. So as you see, we are covering up in equations along with functions. So we don't have to do it separately. So we'll be covering up modulus function. We'll be covering up greatest integer function. Greatest integer function. This function is also sometimes called the GIF function. GIF, the greatest integer function. Many books will also call it as floor function. It's also called floor function. Why does it call floor function? We will discuss about it when we actually reach this function. Then we'll talk about the least integer function. Least integer function. Least integer function. LIF, also called as ceiling function. So this is also called as the ceiling function. Why does it call ceiling function? That we will talk about when we start discussing this type of function. And finally, we are going to talk about fractional part function. Okay, fractional part function. And last but not the least, we'll take up few miscellaneous cases of functions like max min functions. So lately in competitive exams, they have started asking max min functions also. So what are they? We will discuss about it when we reach that particular part. Okay, so primarily your entire domain and range will be covered for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Okay, so almost there are 10 functions, 10 types of functions which will be addressing. So there is one more. What is that? Mahesh, anything that you feel I have missed? Okay, signal function. Okay, signal function basically I would try to cover in this particular part which is called piecewise functions. Okay, so anyways, we will discuss about piecewise functions. And we'll talk about signal function there. Okay, piecewise functions. Is it fine? So primarily, we will be covering these type of functions in our discussion. Okay, so algebraic function, transcendental functions and special functions. Now please understand, we will be covering these functions from our domain range point of view. We are not going to do PSD in these particular functions. Right, so they are just the functions that you will come across. And hence you should be aware of the concept of domain and range and a bit of graph. Okay, so that you don't have a problem addressing any questions which come in association with these functions, even in class 12th. Okay, so I hope you have taken a note of this. Shall we now start with polynomials. Good afternoon to everybody who has joined in. I hope you all had your lunch. I think your classes got over by 1220, right? Did you get time to have your lunch? No, okay, maybe in the break you can have it. Okay, so 40 minutes of break was there, so I thought you would have your lunch during that time. Okay, anyways, let's get started. So I'll be starting with polynomials now. So guys and girls, you have been already been introduced to polynomials in class 10th, right? So I'll be going slightly faster and we will not do PSD on this topic. Let's not spend too much time on this. There is a topic called theory of equations where we'll be dealing with this topic even more in detail. But primarily our focus is just to find domain and range and just a bit of know how about the graph of these functions, right? So we will not devote too much time on this. Okay, so what are polynomials? Polynomials are basically functions of this type. Okay, these are called polynomials. Okay, and you already know that in a polynomial, the variable involved, whatever you want to call it x, y, t, the variable involved, or the exponent of the variable involved, exponent or you call power of the variable involved is always a whole number, right? So this is the structure of a polynomial and the exponent or the power of the variable, whatever is the variable using the polynomial, that must always be whole number. It cannot be fractions. It cannot be negative integers. Okay, so those are called polynomials. So let me test your understanding of polynomials. Is this a polynomial? Let's say I call f1x, x cube minus 3x squared plus 2. Is this a polynomial? What do you think? Yes or no? Yes, it's a polynomial. Okay, what about this? 1 by x square plus 2x plus 1 or 2x plus 5, is this a polynomial? Is this a polynomial? No, it is not a polynomial because the structure of this doesn't match with this structure. So both should have, this should have the structure like this. So in this, I have not written something reciprocal of something, right? So this is actually a rational function, but it is not a polynomial. We will talk about rational function in some time, not too early. Okay, try out this one. Is this a polynomial? x to the power 3 by 2 minus x plus 5. No, it is not a polynomial because there is a fractional power sitting over there. Okay, so I hope everybody's clear about the polynomial concept. Okay, now this coefficient is what we call as the leading coefficient. Okay, leading coefficient. And if your degree of the polynomial is n, then please note that the leading coefficient cannot be zero. Right, so the degree of the polynomial is n, then the leading coefficient cannot be zero. Now this leading coefficient is a very vital element for a polynomial because first of all, it is leading, it is attached with the term of x which has got the highest power. Right, so it takes a lot of, it makes a lot of difference. When it comes to the graph because of this leading polynomial, so leading polynomial plays a very vital role in deciding which way the graph is going to be led. Right, in which way the graph of the polynomial is going to move. Okay, so the leading coefficient is very, very important when it comes to making the graph of that polynomial. We'll see how we'll see how. Now again, let us not lose target. What are we studying here? We are studying domain and range. Right. So what I will do is I will start picking up polynomials one by one and we'll start talking about its domain and range. So what is the very, very, very, very, very basic polynomial that comes in your mind? A constant, isn't it? A constant polynomial is the lowest degree polynomial that you can actually come across. Right, so a constant polynomial is a degree zero polynomial. Right, something like f of x equal to c. Okay, where c is some real number. Okay, so this is the basic, this is the degree zero polynomial provided c is not zero. Okay, now this is very important. If c is zero, okay, then we say degree is undefined. Okay, degree is undefined. So please note zero as a polynomial, the degree is undefined. Right, and if c is not equal to zero, then your degree is considered to be zero. The degree is zero in this case. Okay, please note this down. So let's see the graph of this. Those who were in the bridge course, you would have already come across the graph of this. So I will not take your time much. I'll be moving a little faster. So if you see the graph of a constant polynomial, please understand, no matter whatever x you feed, the output will always be a constant like this. Right, so this is how a graph of a constant polynomial will look like. So no matter whatever the input, let's say if you put one, the input will be c. Right, if you put a two, the input will still be c. If you put a three, the input will still be c. If you put a minus one, the input will still be c. Okay, so this graph is going to be a flat line, a horizontal line. Even if c is one, degree is zero also this. Even if c is one, degree is zero. Because if you want to write one, you can write one as one x to the power zero. Yes or no? So this will be considered to be a zero degree polynomial. But if you have a zero, then what happens? Zero, if you write x to the power anything, it will be a zero. So you can have a degree of one, zero, two, three, four, five. One raise to anything is one. Yes, but how is it, what has it to do with the degree of the polynomial? So then one degree of one should also be undefined by that logic. No, no, no, no, no, no. So in this site, degree is the power of the variable involved in that expression. This is acting as a coefficient. This is the variable. This power is, the power of the variable is what we are worried about as degree, not the power of one. Sir, f of x equal to c, that one sir. So in that one. Yes, so here the variable will be considered to be zero, x to the power zero. So then it could be written as zero x zero also, then also the degree of zero is defined. Also there is a sentence. Zero x zero can be written, zero x one also can be written, zero x hundred can be also written. Does it make any difference to the answer? No sir. That's why the degree is undefined. Because there are so many degrees possible. Got it sir, got it. Right, but when you are writing a one, you can always only write, you can only write one x to the power zero, nothing else. If you write one x to the power one, it will become an x. Oh yes sir, yes sir, got it. Got the point. Yeah. All right. Now look at the graph and tell me what is the domain and range. Now I think Subra Sir would have already told you that for domain of the function, we always see the span of the function along the x axis. Right. So note this down. Note this down for domain. Okay. You see the span of the function along the x axis. That means how spread is that function along the x axis. And for range of the function, you see the span of the function along the y axis. All right. So if you look at this graph, what is the span of this function along the x axis. You'll see that it is spreading across the entire x axis, isn't it? So in this case, your domain will be all real numbers or you can say minus infinity to infinity. Both are considered to be the same thing. But what will be the range? What will be the range? Who will tell me the range? Range will be a singleton set. Right. Range will be a singleton set C. Okay. So any constant polynomial, the domain is all real numbers. The exhaustive domain is all real numbers. And the range is that value of, you know, the constant, which is taken by the function for any input that you're putting inside. Okay. So whatever input you put, the output is always going to be C. Right. So C becomes your asset having a C will become your range of the function. Okay. So this is to be noted because this is what we are going to take away from this entire exercise. Okay. Please note this down. Next, we are moving to move towards linear polynomial. So I'm not going to spend too much time. What did I say? My main agenda is to address domain and range. We are not here to do PSD on these topics. Okay. Of course, we'll have ample time to do all that, but not right now. Any questions so far, anybody related to constant polynomial. And please remember this while you are seeing the graph. If you have been given a graph of a function and you want to see its domain, check its span along x-axis. Okay. And if you want to see its range, check the span of the function along the y-axis. Span means everybody understand the spread, the spread of the function along x and the y-axis. So what if it was f y? So that's a function of y. So it's a polynomial in y. So your y will be along x-axis and f y will be along y-axis. Sir, then it will be span of f y along y-axis. Will it be changing the domain and range? My dear x and y are just name of the variables. Okay. So whether you draw a graph of y versus f y or whether you draw x versus f x. So whatever is the span of the function in this direction, that will be your domain. So this is your input span. So whether you call that input as y or z or a t or a u, it is up to you. So the spread of the function along the input axis. That means what you can put inside the function. Those set of values are called the domain. So I'm saying that if you take the input axis as the vertical axis, then how would it be? How would it look like? See, and then whatever input axis you are defining, that itself, the set of all values sitting on that axis, they will become your domain. Even if it's the vertical axis. Whatever the vertical axis, but the convention says that we always take the dependent variable on y-axis and independent variable on x-axis. So we follow the convention that if the input or the independent variable is on the x-axis, the span of the function along the x-axis will be considered to be the domain. So the idea is input, whichever axis gives you the input or whichever axis is your independent variable means something which doesn't depend on anything. You are just putting it without any thought about it. That axis will contain your domain and the output that comes out from that input, whichever axis you are using for it, that axis elements will be called the range. Is it fine? All right. So with this, we are now going to head towards linear polynomial, linear polynomial functions to be more precise, linear polynomial functions. So what is linear polynomial functions? Quickly speaking about it, so a function of this type is called a linear polynomial function. Of course, a should not be zero. If a is equal to zero, then it will actually become a case of constant polynomial. So if you talk about linear polynomial, even though that constant polynomial is a line, not to get confused with a linear polynomial whose graph is also a line. Don't get confused. Constant polynomial graph is also a line. Linear polynomial function graph, that will also be a line. But let's not get confused between a constant and a linear polynomial. The nomenclature uses different for both of them. Now I think you have all done this in your autonomous course that a is called the slope of the line. So whichever line comes out from the graph, a is called the slope. What is slope? Anybody know slope? It's a ratio of rise to run as you move between any two points on the line. Very correct, Rishabh. So let's say this is a line. Okay. This is a line. And I take any two points on it. Yeah, that is an expression for a tan theta. But primarily I wanted to know whether you know this fact. Let's say if you're moving from a to b direction. Okay. So this is called the run and this is called the rise. So this is called rise and this is called the run. Okay. Please note rise and run could be negative quantities also. For example, if you're going from b to a rise will be negative because you're actually going down. So the ratio of these two quantities is what we call as the slope. Okay. Now, please note that in our linear polynomial, we are not going to talk about such lines whose slope is zero. Okay. Even though they are zero slope lines also. But in linear polynomials, we are not going to talk about that because a cannot be zero for us. If a is zero, it will become a case of constant polynomial function. What is b on the other hand? B on the other hand is called the y intercept. Okay. You're already aware of it. What is y intercept? Y intercept is basically where the line will cut the y axis. So let us say I just saw my coordinate axes over here. Let's say this is my x and the y axis. Right. Now this length, this length is what we call as the y intercept. This length is what we call as the y intercept. Now, y intercept is always calculated from origin. So if you're going down, so to calculate the distance, if you're going down, then that y intercept will be negative. If you're going up, then the y intercept will be considered to be positive. Right. So here in the present case, since the line is cutting the y axis below the origin, in this case, our y intercept will be negative. Okay. Hi, Pratik. No worries. Pratik, we were doing the concept of domain and range of polynomial functions. Okay. So I'm into my second function right now. So we have already covered a constant function over here. Okay. Don't worry. You have not missed out much. After the class, when you see the notes, you'll be able to understand everything. And of course, the recording will also be shared. Okay. Why intercept the distance or the point? Yeah. See, when you say this point is 0 comma B, then B automatically signifies this directed distance, my dear. Why does it call directed distance because B can be positive or B can be negative. Another way to understand Vasudev is it is the coordinate of the point where the line cuts the y axis. You can treat it like that also. Okay. Anyway, it is convenient to you. B is nothing but the directed distance from the origin to the point where the line cuts the y axis. That is one way of understanding it. Another way of understanding it is it is the ordinate or it is the y coordinate of the point where the line meets the y axis. Is that fine? Any questions? Okay. Directed. In this case, since you're going down, the directed distance will be negative. If you're going up, let's say my line was like this. I'll take another example of a line. Let's say my line was like this. Okay. In this case, your directed distance will be positive. So this will be considered to be positive. Right. In this case, it will be considered to be negative. Correct. Make sense. Okay. Now quickly tell me what is the domain and range. I saw one more thing. Many people ask me, sir, can I have a infinitely big? If your slope is infinitely big, please understand in that case, your line will become a vertical line. So let me just show you two, two situations over here. If your A is zero, your line is going to be a horizontal line. So this is when your A is zero. Okay. So we will not be talking about this. We will not be considering this to be a linear polynomial function. So this is not a linear polynomial function. Okay. It's a constant polynomial function. Right. Okay. And if your line is vertical like this. Okay. Even though it is a line, but note that this is not a function itself. This is not a function itself. Why? Because functions, if you draw a vertical line, the function will be cut only at one point. Isn't it? So here if I draw a vertical line exactly overlapping with this line, it is going to cut it at so many places. Right. So in two cases, we are not going to address. Okay. Because this is already addressed, only address under constant polynomial. Okay. And the second one is not even a function. So why to, why to talk about its domain and range? It is a relation, but it is not a function. Right. Second one is not a function because Vasudev, I think you would have learned the vertical line test. Was it taught vertical line test? So, I would have definitely mentioned about it. Vertical line test. No. Yes or no. Anybody, can you please? He did. Okay. So vertical line test says that if in any relation, you draw a vertical line. Okay. Let's say I draw a vertical line like this. Okay. And if it cuts the graph at more than one point, then it will not be a function. Right. So it fails the vertical line test. Okay. So it's a relation, but it is not a function. Clear? Any questions? Any concerns? Any questions? Any concerns? All good so far? Okay. So, yes. Now look at the graph and tell me what is the span of this graph? Take any one of the graph, whether you take the blue line or the white line. Tell me what do you think will be the domain of this function? And what is the range of this function? Right. So domain will be again set of all real numbers or from minus infinity to infinity. A range will also be all real numbers. That is minus infinity to infinity. Okay. So please make a note of this. This is going to be your domain and range of any linear polynomial. Right. See, we'll talk about this chapter again in our straight lines concept under coordinate geometry. But as of now, my target is only to talk about domain and range. So we will not do a lot of, you can say, probing on this concept till we reach the straight line. So shall we move on to quadratic polynomial of quadratic polynomial? Any questions anybody has? Okay, let's move on. Good. So quadratic polynomial, quadratic polynomial, any polynomial of this nature is called quadratic polynomial. Okay. Now, here most of you would be knowing the role that this leading coefficient plays. Right. So many of you would be already aware that a basically controls the concavity of the graph, the concavity of the graph of the parabola and sorry, concavity of the graph of the quadratic, which is actually a parabola. Right. Most of you are already aware that a quadratic polynomial graph is a parabola. Now how it controls the concavity of the parabola. I would quickly like to show you a demonstration here. Let's see a demonstration on GeoGibra. So what I'm going to do is I'm going to, I'm going to choose a slider here. Okay, slider is basically a tool that we have on GeoGibra by which you can change the values of certain parameters. Right. I begin with a very simple graph, y equal to ax square. Okay. And a I have not mentioned right now because a could be anything between minus five to five. So this tool has automatically chosen a value as one right now. Okay. But I can change it. I can go all the way till five or I can go all the way till minus five. Now just watch what happens to the graph. If I start increasing the value of a. See, as you see, as I increase the value of a, the graph is becoming more thinner and thinner. Right. So this is basically influencing the con cavity of the graph. Later on, you will realize that this is something to do with the double derivative. Okay. As of now, I'm not going to use very technical terms of calculus because you have not done calculus. So later on is basically associated with the double derivative. Okay. And if I go backwards, let's say if I start moving back to, let's say one again. Okay, it's opening up. And if I start going towards fractional values, it is becoming fatter and fatter. Exactly at zero. If you see the equation will become the graph will become flat, just like a line. And if you start going to the negative directions, you can see the function will start opening downwards. Okay. So overall, if you see this is how a will influence the, the parabolas con cavity. Do you see that? Okay. Very important. Now, how does, how does be influence the graph? Let's talk about be. Okay. Now be basically, if you change your be, it basically move makes your parabola move on a parabola. Okay. So if this guy changes, okay. Change of be makes the parabola or makes, makes the parabola parabola move on a parabola. Okay. I'll show you how it, how it happens. Okay. Very interesting thing. Or maybe you would have never noted it down. So what I'm going to do is I'm not going to change my slider. Okay. I'm not going to call it as be this guy. I'm going to erase. Okay. And I will write any parabola x square. Let's say x square plus bx. Okay. And let me have a constant. Okay. So what I have done is I have written a parabola whose a is known is one. I have chosen in this case be something which I can change. Okay. So I have kept a slider for be. So be I want to change and see, I have kept it as minus two. Okay. So you pick up any point on the parabola, any point, let's say I pick up this point. Right. Now just watch the motion of this point when I am changing my be value. You see a is moving on a path which is actually parabolic. Okay. So no matter whatever point you take up, they're all dancing on a parabola. So changing the value of be will make every point on the parabola dance on a parabola. So it will be dancing as if it is on a parabola itself. Okay. Now a lot of things I know are actually asked on this concept in J exam. One, one question that can be framed is what would be the equation of this blue trace itself? Okay. So that question could be possibly asked. But one thing I can tell you is that for the blue parabola which you see on your screen, the a value or the leading coefficient is exactly negative of the leading coefficient of the original parabola. Okay. Some of you are saying, can I show you some other point also? Okay. Let's take this point. Okay. So what's the motion of be again when I'm changing? Yeah. Again, when I'm changing the VC, it is also moving on the same parabola. Right. So every point is going to dance on a parabola. Is it clear? So how does your a influence the graph? How does your B influence the graph? Is that clear to you? Any questions here? Okay. Now what effect does C have to play? C just makes the parabola move up or down. So it is responsible for the upward slash downward movement of the parabola. Downward movement of the parabola. Okay. And please mind your pronunciation. It's not called parabola. It is called parabola. Okay. Parabola parabola is basically a word which was used coined for the path faced by an object thrown under gravity. Okay. So A, B and C, I hope it is now clear. Okay. Let me show you how C is influencing it. So let's, let's reopen the GeoGeballer tool once again because instead of erasing so many things is better. I reopened the entire file. Yeah. So let's have a slider. I'll call this slider as C just because my coefficient name is C. I'll just call it as C. And let's write down the parabola X square plus X plus C. Okay. Now if I move my C here, see how the parabola is going to move. It's just going up and down. Okay. It is not dancing. Just like it was dancing in B like this. In this case, it is just moving up or down. Okay. So I hope the idea behind how A, B and C are influencing the parabola graph should be clear to you now. Is it fine? Okay. Now I can go, I can do this. I can do PSD on this topic, right? But that's not what we are here for. We are here for learning its domain and range. So this topic anyways will be covered under complex numbers and quadratic equations. Okay. So we'll come back again to this. So this is a very big topic for us in class 11. Okay. How can it move up and down and in a parabola at the same time? So the point on the parabola is actually moving up and down some month. Okay. So whatever C is there, by that value of C, it is going to go up or down. Let me show you. See, if you take any point, let's say I take this point. Okay. So just watch the trace of this when I'm changing my C. See, it's just moving up and down. Okay. Any point you take, you can take a point here also. Yeah. So just see the trace of this trace means the path which is taking. Okay. It's moving up and down only. Right. Can you please change all at the same time and show what happens. One by one I think is more observable. How to make how to make it move left and right. So if you want to make it move left and right, you have to make a change to this fellow. Okay. So that also to you, Vasudev, sorry, Mahesh Mahesh Mahesh has a question. How do I make the parabola move up and sorry, right and left and right. Correct. So I'm going to change my X with X plus C. Okay. And I'll see when I'm changing it, it's moving left. Okay. We will talk about all these things in detail, not to worry. Yeah. When I'm taking theory of equations, we will go more into detail of that. See what will happen if you change A, B and C, the parabola will become thin, fat will move also will move up and down. So it will be a chaotic movement. What do you want? What would you like to know about it? Sir, can a parabola open downwards? Yes, my dear. If I change my quadratic equation coefficient to negative, it will open downwards, which I think already I discussed it, right? If I put a minus sign here, the parabola will open downwards. So can't it be analyzed like one at a time? Didn't I do that already? Didn't I show you what will happen when A changes? Didn't I tell you what will happen when B changes? And didn't I also tell you what happens when C changes? So why you want to make everything change at a time? Okay. So if you have so much of interest in knowing it, I will do it for you. Okay. So Vasudev has a very, very special request. He's asking me to change everything. Okay. So Vasudev, let's try it out. It will be a full quichadi, but still if you want to see it, let's do it. So what I'm going to do, I'm going to set up, I'm going to set up three sliders here, A, B and C. Okay. So slider A, let's also set slider B. Okay. And also let's set slider C. Yeah. And now I'm going to write a quadratic, Y is equal to AX square plus BX plus C. Okay. Now I will put everything on. See, it's going up also. It's expanding, contracting also. It is moving also. So nothing you can guess out, right? So you're just making everything change. Is it fine? Any questions here? Just, you wanted to see the animation effect maybe. Okay. All right. So let's stop this and we'll move on to something related to our domain and range. Yeah. So guys, we have already discussed here, guys and girls, we have already discussed here that if A is greater than zero, your graph of the quadratic parabola is going to open upwards. Right. So the graph is going to look like this. Okay. I'm just drawing a rough graph. And if A is negative, if A is negative, your quadratic equation graph is going to open downwards. Now, I'm sure all of you know what is this point called this point or this point? What is it called? It's called a vertex. Right. This is called the vertex of the parabola. Later on, we learn that vertex is basically a point or point of intersection of the axis of the parabola with the parabola. This is called the axis of the parabola. What is the axis of a parabola? A line which cuts the parabola into two symmetrical halves that is called the axis of the parabola. So this is what we call as axis. Okay. Axis. Now, what is the coordinate of the vertex of the parabola? Coordinate of the vertex of a parabola. This coordinate is given by minus B by 2A comma minus D by 4A. Where what is D? D is the discriminant. I hope you all have heard this term discriminant. What is discriminant? B square minus 4AC. Why does it call discriminant? Why does it call discriminant? Because it discriminates between the nature of the roots. Correct. Now, please note that the vertex coordinate plays a very vital role in deciding the domain and range of the graph. In fact, very vital role in deciding the range of the quadratic polynomial. Okay. How we will see in some time. Many people ask me, sir, even if the parabola opens upwards or downwards, does the vertex coordinate remain the same? Yes, my dear. The vertex coordinate always remains minus B by 2A comma minus D by 4A for an upward or downward opening parabola. Mind you, the parabolas can also open left or right, but those will not be considered as functions. So we will be not talking about those parabolas unless until we are into conic sections chapter. Okay. Now, look at the graph and tell me whether A is positive or whether A is negative. What should be the domain of the function? What should be the domain of the function? All real numbers, right? Because the graph is spreading in both directions indefinitely, right? So these arms are not going to stop. So these are going to go to plus infinity on the right side and minus infinity on the left side. So if you look at the span of the function, the span of the function will be the entire real number line along the x-axis, right? What about the range? What about the range? What about the range? Now, range will depend upon your A value. If A is positive, then what will be your range? Now, look at this left graph. If yes, yes, correct, Rishabh and Mahesh. If you're, no, sorry, not Rishabh, Mahesh, yes. If A is positive, the graph will start from this vertex point and go all the way up to plus infinity, isn't it? So along the y-axis, the graph is spreading from minus D by 4A to plus infinity, correct? Now, need not write plus. If you just write infinity, it is considered to be plus infinity. And if your A is negative, now look at this second graph. So you start from this position and go all the way down to minus infinity, right? So in this case, our range will become from minus infinity open to minus D by 4A closed, okay? So kindly make a note of this because, you know, if you get a direct question on finding the domain and range of quadratic function, then you can use this formula. Is it fine? Any questions? Okay, can we have a small question? Just a small question. For this polynomial function, okay, find number one, the domain of the function, the domain of the function and the range of the function. Quickly, give me a response on the chat box. Domain is easy. Domain you don't have to worry about. What's the domain answer? All real numbers, right? All real numbers, okay? So I'm more worried about the range. I'm more worried about the range. So give me a response for the range. Correct, Sanjana, correct. Mahesh, just a quick advice from my side to everybody. If you're mentioning any interval, okay, let's say any interval you are mentioning, A to B, A should always be lesser than B. So the least value should always be on the left side. More value should be on the right side, okay? So never say something, somebody I think wrote minus root 17 by 2 to minus infinity. This is a wrong way to write it because minus infinity is smaller. So that should be on the left side, okay? All right, so let's discuss it out. Let's discuss it out. So in this case, since your A is negative, since your A, that is a leading coefficient, is negative. The graph is going to be opening downwards, right? So your range is going to be minus infinity to minus D by 4A, right? So your answer will be minus infinity to minus D. Now minus D is 4AC, minus B square. Or let's calculate D separately. So D is B square minus 4AC. So B square is 9. Minus 4AC is 8, correct? So D is 17. So this is going to be minus 17 by minus 8. Or you can write it in a more resolved form. 17 by 8, okay? So this is going to be your answer for the range. Most of you got it correct, well done. Most of you got this correct. Is it fine? All right. So as I told you, we are not going to do PSD on quadratic, right? This anyways is a topic for you in not only complex number and quadratic equation chapter, but it's also a chapter in the conic section. So we'll get ample time to do a lot of, you know, going into much depth in this particular concept, when we reach those topics. So immediately my main focus would be to be addressing the domain and range. So I'll move on now to cubic polynomials. So if you have any questions related to domain and range per se of quadratic polynomials, please feel free to ask questions. Please feel free to ask, you know, stop me and, you know, ask your doubts. Okay. If no doubt, we can move on. All right. So we'll now move on to cubic polynomials. Now, again, my purpose is not to do PSD on this topic. Okay. So I'll just talk about domain and range. Now, how does the cubic polynomial normally look like? A cubic polynomial, basically equation is a degree three polynomial equation, something of this nature. Okay. And again, this A plays a very vital role in the graph of the cubic polynomial. So if A is positive, if A is positive, roughly speaking, roughly speaking, the graph of a cubic polynomial looks like this. I mean, this is a rough estimation of the graph. Okay. Please don't take it as if it is always going to cut the x-axis at three points. No, not necessarily. Many times it will just look like this also. It can be like this also. Okay. Sometimes I've seen the graph will be like this also. Okay. So it is just a rough estimate that I'm drawing on your screen right now. Rough means a very generic view. It's like, you know, I'm just trying to show you a lion. Right. Now, if you go to different parts of the world, lions will be different in shapes and sizes. Right. So for example, if you go to the savannahs, the lions will be the big and strong. And our Indian lions are like, like this. So my purpose is not to exactly represent every lion, but to show how a lion typically looks like. Isn't it? So in the same way, I'm trying to express how a cubic polynomial actually typically looks like. So what is more important here is to see where are the end branches of this polynomial? So if A is positive, your graph will always end towards plus infinity if your x becomes very, very large. And the graph will always end towards minus infinity if your x is negatively very, very large. This is very important because from this, we'll get an idea about its domain and range. Okay. So again, note this down. We are going to talk more about it under theory of equations. Similarly, if your A is positive or negative, then your graph will roughly, roughly again, I'm not drawing a very accurate graph for all situations. Your graph will actually look like this. In other words, your graph will be going towards minus infinity if your x is going towards plus infinity. And your graph will be going towards plus infinity if your x is going towards minus infinity. Okay. So these are the two points, two arms that is worth noting down because again, we will come to know about its domain and range from that. Now, many people ask me, sir, why do you think it's going to go up to plus infinity when x is very large and minus infinity when x is very negatively very large? See, very simple. If you keep this as positive. Okay. And please note, this term will be the dominating term out of these four terms because here the x is having the largest power on it. So if you go, if your x is becoming very, very large, then this whole thing will be positive. So your graph will go towards plus infinity. But if your x is going towards a very, very negatively large value, this whole thing will become negative, negative, very, very large. So your graph will go towards negative infinity. And opposite is going to happen when your A has the negative sign. That is the reason why the graph looks like this. Okay. Yes, Veehan. I'll show you on the, on the GOG graph. See, let's, let's have a quick demonstration of these graphs. So I'm just opening a graph. Yeah. So let me draw a cubic polynomial graph with a positive coefficient. So Y is equal to 2x cube, 2x cube. Okay. Minus. Now, if I just stop at 2x cube, if you see, don't worry about the between part, what is happening? Just see the end arms and arms will always be towards plus infinity as you go towards the right side. So this will go to plus infinity as you go towards plus infinity on this side. And it'll always go to minus infinity if you go to minus infinity on this side. Okay. So even if I write just 2x cube, it is going to show you the same trend. But let me now write few more terms to this. Let's say I add a negative, negative x square maybe. Okay. Negative 3x square. Okay. Plus 4x. Okay. Minus 5 maybe. Let's write. Yeah. So again, whatever you write. See, many people, they are worried about, said you have shown some ups and downs here in the graph. What are these ups and downs? Now these ups and downs, we don't have to worry about right now. Okay. By the way, this is called the local maxima. This is called the local minima. So I'm not worried about local maxima, local minima because those are subject matter of calculus. And those are anyways not going to decide our domain and range. So our main focus is domain and range only. Right. So what I want you to see is that where are the N arms going? Where are the N arms going? This is more important to me. Okay. Similarly, if I have a negative coefficient of x cube. So if I just make it negative 2, then see what happens. Okay. So as you can see here, the graph has now shown this kind of a nature, the one which I've shown on this side. You see that? Okay. So what is more important is where are these N arms going? Where are these N arms of the graph going? That is more important to me right now. Clear. Okay. So these are the local maxima, the local minima and they require calculus. So I don't want to comment on them right now. So they need calculus to be addressed and calculus is something which we have not yet started with, but we'll be starting very soon towards the end of the year. So looking at these two graphs, let us come back to our main agenda. What is the domain? What is the domain here? And what is the range here? What is the domain and what are the range here? Look at the span of the function along x axis. Domain is all real numbers, right? Range is also all real numbers because it is going to extend in both direction along x axis and both directions along the y axis. Okay. Yes, Anirudh, it is going to be shared, but don't worry about missing whatever you have missed so far. Take a particular polynomial as a new concept. Okay. So don't worry, don't think like, okay, since I missed the first part of the class, I will not be able to understand anything, nothing like that. Okay. So just start, every polynomial is a fresh function for us. Okay. So we'll be talking about domain and range of those polynomials freshly. It has nothing to do with the previous polynomial. Fine. So pay attention from here also. Okay. You have not missed much thing. Is it fine? Again, may our main focus was domain and range that we have already done. So we will now move on to the next polynomial. So you must be wondering, sir, how many polynomials are you going to cover? Because there are infinitely many polynomials existing. See, I'll only talk about bi-quadratic and after that we will summarize. Okay. After that we will summarize. So let's move on to bi-quadratic polynomial. Bi-quadratic polynomial. Bi-quadratic polynomial. Okay. How does a bi-quadratic polynomial look like? So it's a fourth degree polynomial equation. So something like this is a bi-quadratic polynomial. Okay. Now, again, this A has a very vital role to play in deciding the graph. So if A is positive, a bi-quadratic polynomial graph will roughly, roughly, again, mind you, my dear, I'm using the word roughly, not exactly. Okay. Every time it all depends on A, B, C, D, E and E. Right. But what I'm showing you is a generic picture of a bi-quadratic graph. So it will look like this. Okay. So A is drawing a rough version. So what is important for us is these two arms of the graph. We are not going to worry about these local maximas and local minimas over here. Okay. That is a subject matter of calculus. What I am more interested in is the two ends of these graphs. Okay. Because that is going to help us know its domain and range. And if A is negative, the graph will show a reverse direction trend. So it will be somewhat like this. Okay. I'm just drawing roughly. I'm just drawing roughly. Okay. So it is going to go down. So both the arms are going to go towards negative infinity. So in this case, when your A is positive, as you increase the value of X to plus infinity, this graph will go to plus infinity. And if you also decrease the value to minus infinity, it will still go to plus infinity. Okay. And when your A is negative, the other way round trend will be seen. If you increase it to plus infinity, the graph is going to go down to minus infinity. And this also, if you increase to minus infinity, the graph is going down to minus infinity. Okay. And the reason is the same. The reason is the same because this is the dominating term. And note that even for plus or minus, this is always going to be positive. So if this is positive, everything will open upwards. And if this is negative, everything will open downwards. Okay. So that is how your graph offer by quadratic is going to look like now. Who will tell me the domain of this function? Quickly. What is the span of the function along X axis? All real numbers. Right. And what are the range of the function? Now, please understand here for range of this function, we actually need calculus. Okay. Calculus is needed. Let me call it as calculus. Calculus is required or needed. Okay. Why calculus is needed? Because see, if you talk about this case, here we will be basically starting from this position, whatever is this position and we'll going all the way up to plus infinity. Right. But how do I know this position? How do I know this position? So this position will require you to use calculus. Now, that is not a vertex. This is not a vertex that you use your vertex formula for quadratic and get the Y coordinate of this point. No. To know the Y coordinate of this point, you would have to apply differentiation. Yes, you have to differentiate it. So since you have not been, yes, Mahesh. I feel like for like for quadratic, we found minus b by 2 a that we use that person found out and it's gentle. You say you're trying to say that this is minus b by 2 a or something like that? No. I'm saying that again, take a general, you know, by quadratic polynomial differentiate, define the minimum and just use it as a general formula. My dear Mahesh, you know differentiation. Is there anybody else other than Mahesh and Rishabh who knows how to differentiate? Sir, no sir, even for minus b by 2 a, we didn't know differentiation. Just we mark the formula minus b by 2 a. No, minus b by 2 a doesn't come from differentiation. It comes from the act of completing the square. It can also come from differentiation, but that is not the only way to get it. But in a by quadratic, without differentiation, you cannot find the coordinates. Yes, sir. Got the point what I'm trying to say. Yes, sir. In quadratic differentiation was not necessary to get that coordinate of the vertex. But in by quadratic, if you want to know this minimal location, you have to differentiate it, put to zero, solve that cubic equation, then do a double derivative test, all those things will come into picture. But other than three of you, maybe Pratik, Rishabh and whatever, however, many people don't know differentiation. Sorry, we already did it in physics. You already did it in physics? Everybody knows? Yes, sir, we did it in physics. Okay. So if that is the case, then of course when it comes, you apply differentiation and get your results. Okay. Many of you. We didn't do maximum minima in physics. Huh? We didn't do maximum minima in physics. Yes, that's what many of you are saying I don't know in full depth, right? Maybe I've just skimmed through the topic. Okay. So we will discuss it, not to worry. So meanwhile, when the concept of calculus is not being covered, we will not take up questions on the range for bi-quadratic. In fact, for the subsequent polynomials, we will talk about it in some time. We will not be talking about the domain and range. So even in this case, you don't know the coordinate of this point, right? So this requires calculus. So without that, I cannot comment about its range because I would need calculus for it. Okay. So we'll wait for the right time to come. We'll wait for the right time to come to take up the range concept of polynomials. If you take many bi-quadratics, plot them, you will get a general pattern with which you can get the nine others. No, that doesn't work. That doesn't work possibly. Okay. Every, you know, bi-quadratics can go anywhere from anywhere to anywhere, right? There's no range, there's no pattern in the range, right? It all depends upon your A, B, C, D, and E values. Okay. So this is not something which you can do by looking at the pattern. Okay. All right. So as I told you, we will not go into too much of depth of polynomials. So we'll try to summarize it, right? So we'll try to take up a summary of domain and range of polynomials. Domain and range of polynomials. So first thing that you would have all noticed that no matter whatever is the polynomial, whether it is a constant, linear, quadratic, cubic, bi-quadratic, this domain will always be all real numbers. Okay. So this is to be noted down. Any polynomial is given to you. No matter whatever is the degree. Okay. Degree 0 to degree infinity, whatever. It's range will always be all real numbers. That means a polynomial graph will always span across the entire x-axis. Okay. Now what about range? For range, you basically have to look at the degree of the polynomial. So if it is an odd-degree polynomial, so let's say I divide the polynomial into two parts. One is an odd-degree polynomial. Okay. And other is an even-degree polynomial. Even-degree polynomial. So for any odd-degree polynomial, your range will always be all real numbers. Right. So whether it is degree 1, degree 3, degree 5, degree 7, degree 9, degree 11, whatever odd-degree polynomials are there, its range will always be all real numbers, which is from minus infinity to infinity. Okay. For even-degree polynomial, if you have a degree 0, okay, if the degree is 0, then you can say the range is a singleton set. Okay. Your range is going to be a singleton set. If it is degree 2, if it is degree 2, here also you can predict. That depends upon a value. If a is greater than 0, then it is from minus d by 4a up till infinity. And if a is less than 0, then its range is given from minus infinity to minus d by 4a. Correct. And for any other higher even-degrees, so let me write it like this. So higher even-degrees, you would require calculus is needed. So we cannot, as of now, comment, but of course when we do calculus, we will talk about it. Okay. So calculus is needed. Okay. So does it mean that we will not get any question related to the range of higher even-degree polynomials? As of now, no, but eventually yes, because anyways you would have done calculus by the time you are writing your comparative exams. Okay. So this is a quick summary of the domain and range of polynomials. Any questions, anybody? No, I will not do that derivation right now, also this, because if I start telling you everything about the quadratic, this chapter is going to take another three classes to complete. I told you, you know, the quadratic equation, vertex, I will derive when we do quadratic equation. Our main agenda is domain range. Let's not talk anything beyond it, because that will just lead us to, you know, go deeper and deeper and deeper into the concept, but it comes very simply by completing the square. Okay. Sir. Yes. Sir, in the center module, the functions, they're just one whole thing, the climate change will combine here. Are you going to separate here or just in one book? Okay. Now a very important thing that I would like to share with you, functions are there in class. 11 and 12 both. Okay. And in your module, they have combined it. Because for the module, it doesn't make any difference whether it is taken by 11th grade or a 12th grade. Okay. So while you're doing the module, you realize that there are many topics that you would not be studying in class 11th, like inverse of a function, composition of a function, even odd functions, periodic functions, type of functions like 11, many one, into, onto, those are all class 12th level topics. Right. So what I would request here is, first you do your basics from R.D. Sharma. Okay. Do your basics from R.D. Sharma. Do your basics from NCRT exemplar. As the time progresses, get, see, get hold of domain and range part of it. Wherever a domain range question is seen, other than inverse, trigonometry and all. So trigonometry part don't be, you know, too much involved. But if you have a rational function, irrational function, if you get exponential function, logarithmic function, or if you have modulus function, or signum function, GIF function, try to address, try to solve questions based on domain and range. Other part you can leave for class 12th. When you reach class 12th, you will automatically come to know about many other concepts. Okay. Function is a huge chapter. It is not going to be covered in class 11. It's going to be also covered in class 12th. Okay. Does this, does that answer your question? Mahesh. Yes. What type of statement is that? R.D. Sharma is more important than center modules for now. There's nothing called important. Some topics, no, no, no. Some topics you will see, you would realize that you should do R.D. Sharma before doing center module. And some topic you will feel that if you do center modules, everything else is covered. Okay. Would you also tell how to solve R.D. Sharma because there's so many questions and solving everything would be like, I don't think it would be efficient enough. See, always solve questions of one type and if let's say the second type question are a repetitive type, you can just skip it. Don't solve everything. Time is not there to solve everything. You must be thinking it's a year-long program. No way. There's so many chapters in the whole year that you will hardly get more than four days per chapter. Okay. So you have to cover so many chapters in the coming year that you can't sit and solve each and every question. So for example, if you see first question you solved, second, third look same, then skip it. I mean, of course, don't skip it under the assumption that you will be solving it. If you feel that, no, I know how to solve it. Then don't attempt it. Okay. Because time is not there to do everything. All right. So with this, we are moving on to rational functions. Rational functions. So what is a rational function? First of all, a rational function is basically a function which is expressed as a polynomial divided by another polynomial. So these two are polynomials at the end of the day. Okay. So many people ask me, sir, even then a polynomial becomes a rational function. Yes. Yes. A polynomial is also a rational function because a polynomial can always be written as that polynomial divided by one. Isn't it? So that also is a rational function. Right. So any function where there's a polynomial divided by another polynomial, that becomes a rational function. Examples could be something like, you know, a constant divided by let's say x plus two. That's a rational function. Okay. Let's say x plus one divided by three x minus seven. It's a rational function. So let's say you have x squared divided by x cube plus two rational function. Okay. So there can be so many such examples cited for rational functions. Now, what are we going to learn here? We are going to learn here two things. Domain and range. Domain and range. Okay. So domain, I will first quickly tell you how do we find the domain of any rational function? Anybody's aware of it? So domain of any rational function. How do you find out? How do you find domain of any rational function? Correct. So it is all real numbers except those values of x which makes the denominator become zero. Right. The denominator should not be zero. So it can take any real number as an input, but you should not put anything which is making the denominator become zero. Getting the point. So note down this formula because this formula is going to be helpful in the entire discussion of rational functions. So whether you talk about linear by quadratic or a constant by linear, whatever type of rational function you generate, this formula for domain is going to be valid. Is it fine? Okay. Now what about range? What about range? So range is something which we will basically look from case to case. Okay. So we will not give you a direct formula for range as we have given as I have given you a direct formula for domain. Range is dependent on case to case. Okay. So we'll take few cases and then we'll try to see how do we actually find the range? Okay. So domain is very easy. For domain, there is a readymade formula. Note this down. And for range, it will depend from case to case. Okay. Now range is slightly difficult. Range is not as simple as finding the domain. Many people do mistakes in finding the range most of the time. All right. So what are the cases? What are the cases we are going to see? So what I have done, I have basically classified this rational function under three cases. Okay. Or three types of questions. So let's start with type one question. Okay. So under type one question, I'll be giving you a rational function which has got a constant polynomial on the top divided by a linear polynomial. Okay. So I am taking a case where my rational function has a constant polynomial on the top and a linear polynomial in the denominator. Okay. Let's take an example of it. As an example, let's say I have taken a polynomial rational rational function as 2 divided by 3x minus 5. Okay. As you can see, this is constant like this one. And this is linear like this one. Okay. Now, let's find domain first. In fact, you tell me what's the domain. I have already given you the formula. All real numbers minus those X which will make the denominator become 0. Correct. Pathik Vasudev Sanjana are very good. Very good. So domain is all real numbers except those values of X for which the denominator becomes a 0. And that value is happened. That value happens to be 5 by 3. So please write it like this. So r minus 5 by 3 is going to be your domain. Okay. What about range? What about range? Now for range, let us look into something very important. What is range? Range is the value of f of X for those X which belong to this interval. Isn't it? So in short, what is range? Range is your output when you feed the inputs to the function. These are all your inputs. Correct. And this is your output. This is your output. So what are your outputs when you feed these inputs to the function? So whenever you are trying to find out the range of any function, always first figure out the inputs. Are you getting my point? So please note this down. This is something which many people overlook. Never ever find the range without finding the domain. Okay. Even if the question says just to find out the range, please always figure out the domain first. Okay. Before finding the range because without the input, there is no existence of the output. So till you know what is the input, how can you comment on the output? Okay. So it's a very, very important practice. Many people say, is it a good practice or is it an important practice? I say it's an important practice. It's a very important practice to figure out the domain first before finding the range. Correct. Okay. So now for the sake of brevity, I will start calling f of x as y. Okay. So I'm trying to find out in what interval will y lie if your x lies in this interval. So if your x lies in this interval, what is the interval in which your y will lie? Very good Pateek and Arisha, we have actually got the answer. So for finding the range, what are the steps involved? Listen to the steps first. The first step is you make x the subject of the formula. Right. Now, why do we do that? I'll tell you the reason behind it. So first I will try to make x the subject of the formula. So if I make x the subject of the formula, what do I get? So from this expression, let's take it to the other side, send the 5y to the other side and divide by 3y. Okay. So basically what I have done, I have made x the subject of the formula. Now, why have I done that? It is because I know that this x belongs to all real numbers minus 5 by 3. Right. That means this guy will also belong to the same interval. And from there, I will get an idea about the interval in which y can lie. Correct. So what I'm using the fact, I'm using the domain of the function. Right. To write the x value like this. And now I'm claiming that this is all real numbers except 5 by 3. Correct. Right. So this can take any real number. This whole thing can take any real number except 5 by 3. Okay. Now, how do I get the value of y from this? Now listen to this very, very carefully. So first of all, if your expression here is always real, what is the value that y cannot take? If you want this to be real first of all. It is real, right? Yeah. What is the value which y cannot take? Zero. Right. So the first thing that I'm going to write is y cannot take a value of zero because if it takes a zero value, this entire expression becomes undefined. Right. And that's not how that's not, that's not what I'm expecting the function to give. Right. So I don't want, I don't want the entire expression here to become undefined. So because it is x value, x is all real. It is, it is not undefined. Okay. Now, I know, I also don't want this expression to become equal to this. Correct. I also don't want my expression to become equal to this. So what I'm going to do is I'm going to actually try to make this equal to this. So that I see, I can see what value of y I need to remove. Are you able to understand what I'm trying to do? See, this whole thing cannot be five by three. Isn't it? So I'm equating it to five by three to check what value of y is not permitted. Understood. Just type a yes. If you have understood what I'm trying to say, since this cannot, this cannot take a value of this. I'm trying to see that if I can, that if I equate it to five by three, what value comes out from there? That value of y, I will also exclude. So zero anyways, I've excluded. I will also exclude that y value, which will come out when I solve this equation. But let us see what happens when we try to solve this equation. So first of all, three, three gets canceled. Take the y to the other side. And you'll end up getting a shocking statement. Two is equal to zero, right? Which is not possible. Right. So what does this try to say is that this anyways cannot take five by three. So this expression, which I have circled in white, that anyways cannot be five by three, cannot be five by three. So from here, I do not get anything to exclude. Correct. So here I cannot exclude anything because this expression, no matter whatever value of why you take, can never become a five by three. So the only value which I need to exclude is zero. So what happens to my range? The range becomes all real numbers excluding zero. Okay. So this is your answer for the range. And this is your answer for the domain. Is it fine? Any question, any concerns related to domain and range? Take your time, understand, ask your questions, get it straight. Because later on, things are going to become more complicated. Okay. So would you like to try one more example? If you have understood this, let's do a more question, one more question. Okay. Let's take another question. Question is, you have a function, seven divided by four minus three X. Okay. Question is, find the domain of the function and find the range of the function. Let's do it. Give me a response on the chat box. So the approach, I've already discussed with you, just apply that approach and tell me the domain and the range quickly, quickly. Correct. There are two pratiks, your pratik ranganath, right? And when I see your group, I see one more pratik, not in the class right now, but pratik tejasvi. So who is pratik ranganath? Pratik ranganath is also in Rajaji Nagar or some different school. There's no pratik ranganath. Okay. Then he may be from a different school. Range. Okay. Same answer for the range. Okay. All right. So let's discuss quickly. We don't have much time here. Domain as I already discussed with you, it is going to be all real numbers except those values, which make four minus three X become zero. Correct. No. So I don't want the denominator to become zero. So this will give me four by three. So domain is all real numbers except four by three. Clear. Okay. What about range? Arrange the process is same. We call it as Y, Y equal to seven minus four by three, four minus three X. Then what do we try to do? We try to make X the subject of the formula. Correct. So when you try to make X the subject of the formula, this is what you're going to see. Four Y minus seven is three X, Y. So X is going to be four Y minus seven by three Y. Correct. Now you want this to be all real numbers except four by three. So first of all, if you want it to be all real numbers, Y cannot be zero. Okay. And at the same time, Y cannot take such a value for this, this will become four by three. So first what I'll do, I'll try to equate it to four by three and try to see any value comes out from there. No, I'm disappointed. No value comes out from there. So this is the only, so this is not possible. So this is the only value that you need to exclude. So your range in this case is going to be all real numbers except zero. Okay. Now a question would appear in somebody's mind that sir, can we generalize this? Can we generalize this? So if I have a function of this type A by B X plus C, is there any direct formula to get its domain and range? Yes. Domain will be straight away. All real values except, except, except, except, except, except tell me, tell me on the chat box. Except. What should I write? Minus C by B, right? And range and range will always be all real numbers except zero. So you can actually save your time. If at all a question like this comes to you to solve. Okay. But I would request not to learn this as a formula. Don't try to make unnecessary formulas, which you will not be able to remember for long. Okay. So keep this in mind in case you want to save your time, but don't try to remember this as a formula. By virtue of practice, if you happen to remember it, then well and good, that's a bonus for you. But don't sit and mug this up. Is it fine? Any questions? All right. So we'll now move on to, we will now move on to type two questions. Okay. So in type two questions, we are going to talk about linear by linear. So linear polynomial by another linear polynomial. Okay. Let's take an example. Let's say I have a function which is something like this. 3x plus four upon 2x minus three. Okay. So I want to find out the domain and I want to find the range of this one. Now, as I only told you domain approach is plain and simple. Right. Domain, there is no brainer actually. So domain formula is all real numbers except those x values for which your denominator becomes zero. And your denominator will become zero at three by two. So just remove that value from the domain. Clear. No, no doubt about domain. Anybody? Okay. A range also the process is more or less the same. So what I'll do, I'll call this as a Y. Okay. And I will try to make X the subject of the formula. So let's, let's multiply the denominator to the left side. Okay. So I'm trying to make X the subject of the formula. So let me bring the three X to the left. And let me bring the three Y to the right. Okay. So take X common from here. It'll be two Y minus three. Okay. And X will become three Y plus four upon two Y minus three. So far, so good. Anybody having any problem in the simplification part? I write Richard, I'll come to that. I'll come to that. Your answer is absolutely right. Okay. But let us go slowly because many of the students, they would like to keep it slow and steady. Any, any problems so far. Now, this should belong to the domain of the function. And what's the domain of the function? R minus three by two. Okay. Anything I have missed on, please let me know. Yeah. So this should be R minus three by two. Now, please note, if you want this to be real number, if you want this guy to be real number, what is the first thing that you should, you know, bring into your notice? What value Y cannot take? Can this guy be zero ever? You'll say no sir, that cannot be zero. So the first thing that you need to disallow, disallow Y to take is three by two. So Y cannot be three by two. Because if Y is three by two, this denominator will become a zero. Right. Right. Please don't, don't exclude zero. Zero can be taken by Y. But Y cannot take such a value for which this term, Y minus three becomes zero. Getting my point. And at the same time, Y cannot take such a value for which three Y plus four by two, Y minus three becomes a three by two. So what I'll do, I'll purposely assign it three by two and see, is there any Y value coming up so that I can exclude that Y value. So let's check. So if I cross multiply, this becomes six Y plus eight equal to six Y minus nine. Oh my God. This is not possible because it gives me eight equal to minus nine. So the only value of Y which you need to exclude is three by two. So Risha, your answer is absolutely, absolutely right. So your range will be all real numbers except three by two. Is that fine? Any questions? So domain and range happened to be the same here. Isn't it? Domain and range happened to be actually the same coincidentally, coincidentally. Understood. Any questions? Any questions here? Shall we move on? Would you like to take one more question? Okay. Let's take another question. So question is, if your function is two X minus seven upon four minus five X. Okay. Find number one, the domain of the function and find number two, the range of the function. Okay. Okay. Very good. So I think domain people are taking no time to answer this. Yes. It's quick time to answer the domain. And after this problem, I'll tell you a quick way to find range also. But again, I'm not, I'm not basically trying to give you formulas. That's, that's has never been my way of teaching. So I don't believe in giving formulas because if you just learn formula and go, things can change at no time in the, you know, examination hall and you will be, you know, clueless how to proceed. Okay. So let us understand the concept from the very core of it. So from very depth of it. So that no matter whatever situation comes, you will come out, you know, solving the problem correctly. Very good. Rishabh. Rishabh, do you see a pattern in the range by the way? Just a question. If you can then do let me know. No Pratik range is slightly long, wrong. Just check. Yeah. Take your time. Take your time. Take a minute. Take a minute or two. Right. Mahesh. Mahesh has figured out a pattern. Well done. Okay. Let's discuss this out. Let's discuss this out. So domain as I already discussed, it will be all real numbers except those X values, which will make your denominator become zero. Right. So your denominator will become zero when X becomes four by five. Correct Rishabh. That is the pattern. Very good. And what about range? What about range? For range as I already told you the approach. Okay. Equated to correct Sanjana. Okay. Haa. So the correct. Right. So let's let's discuss it. So now here I'll try to make X the subject of the formula. So let's take the denominator to the right side. Okay. Let's bring X terms to one side. Okay. So X will become four Y plus seven divided by two plus five. Okay. Now this has to be your domain because this is actually your X value. Right. So first of all, if you want this to be real, if you want this to be real, you don't want your denominator to become zero. Right. So this thing should never be zero. That means why can never be two by five minus two by five. Okay. So this is number one. Second thing that you have to look into is you don't want this expression to ever take up value of ever take up value of let's say if I put it as a value of four, four by five, see what will happen if I put it as a value of four by five, then 20 Y plus 35 will be equal to 20 Y plus eight. That means 25 will become equal to eight, which is anyways not possible. So this guy can never become four by five. So this is the only condition that you need to honor. Okay. So your range is going to be all real numbers except minus two by five. This is your answer for the range. Now, for those who are very, very correct, Sanjana. So for those who might ask, what is the pattern for getting the range? It is just the coefficient of X on the numerator divided by coefficient of X on the denominator. That value has to be excluded. So R minus two divided by minus five or for that matter, R minus minus two by five becomes your, becomes your range. Okay. Again, so it's a formula which you can actually use if you want to save your time, but don't wrongly apply one formula to the other. Okay. So in general, let's generalize it. If you have a function of this type, AX plus B by CX plus D, then your domain always becomes all real numbers except minus D by C and a range will become all real numbers except A by C. Okay. So in case you want to remember it as a formula, you may use it, but don't overload your memory. If you are a person who cannot memorize, then don't try using memory to solve it. You will end up losing those sure shot four marks, which you could have any ways got by solving it in a proper way. Okay. Is this fine? Any questions? Any concerns? Okay. So before moving on to type three, now time to take up some inequalities because without inequalities, we will not be able to proceed with the further types of rational functions. In fact, even after rational functions where we are going to start with irrational functions, knowing certain inequalities becomes very, very important. So I will take a pause here from our domain and range of rational function. And in between, I will introduce a prerequisite, which is your inequalities, which is your inequalities. Okay. So we will solve certain inequalities. So we will start with, we will start with a very important mechanism or a scheme that we use to solve a rational function inequalities. Okay. And that, that method is called the wavy curve. Sign scheme. Okay. Some books will also call it as method of intervals. Okay. So this wavy curve sign scheme, is it a scheme given by Modi government? Not really. It's not a scheme of Modi government. It's a scheme where we are going to learn how to, so this method will be helpful. So this is the method to solve inequalities, inequalities, or you can say, inequalities of the below type. Okay. So please listen to the structure of this equation. So I'm just writing a very, you know, ugly looking expression. So it basically helps you to solve inequalities of this nature. Okay. I'm just writing a very generic structure of it. Don't get scared by the structure. Beta 2 to the power of 2, da, da, da, da. Let's say beta m to the power. Okay. So either greater than or less than or greater than equal to or less than equal to zero. Okay. Zero should be on this side. Right. So we are going to learn how to solve inequalities of this nature. This is basically called a rational function inequality. So this is actually a rational function inequality. Okay. Or rational function in equation. Why does it call a rational function in equation? Because if you see this, this term, this is actually a rational function. Right. And inequality because you are learning how to solve inequalities. Okay. Now many people ask me this question. So is it necessary for this zero to be there on the right side? Yes. So this method, which I'm going to discuss with you that can only be applied if you have a zero on the right side. So on the left hand side, you should have any rational function which can include polynomials also. And this could be any of these four inequalities. So either you can have these pure inequalities or these impure inequalities. But on the right hand side, we must definitely have a zero. You must definitely have a zero. Then only this method actually makes sense. Or this method only will work then. Okay. Now while explaining you the method, I will also tell you why this method actually works. I didn't get that. Sir, what if you transfer non zero to other side and take LCM? Yeah. Yeah. Of course. If you had a non zero number, then you can send it to the other side. Okay. And produce a zero. Okay. But right now I have basically taken an expression where you have done all those, all those activities and you have attained the stage. Okay. So if there was a non zero term, you please take it to one side, keep zero on the right side. Right now the expression that you see is after doing that step. Right. Okay. So the process starts from here. That is what I basically am trying to tell you. So what is the process here? Listen to this carefully. Listen to this carefully because many a times, you don't get a clear picture of it in school. Okay. So the first thing that you're going to do here is you are going to make a number line. Okay. By the way, let's take an example to understand this rather than me writing this alpha one, alpha two, those ugly numbers. Let's take an example. Let's take an example. I'll put an example here. So let's say I want to solve this inequality x minus one to the part two x minus three to the power of five x plus one to the power of let's say one divided by let's say x to the power of six x minus two to the power of five. Let's take a different number five. I already taken the six three and express two to the power of let's say one again. Okay. And let me choose an inequality. Let's say greater than equal to zero. Okay. So what I've done, I've taken one example question. Okay. But don't worry from this example, you should be able to get the process very, very easily. Okay. So pay attention. Don't write anything as of now. Put your pens down and just lend me your ears for some time. Okay. Everybody has put your pen down. Okay. Now make a real number line first of all. Okay. So as you see on your screen, I have made a gray color real number line. Okay. On these, on this real number line, you mark the zeros of all these factors that you see in this rational function. So what is the zero of this game? What is the zero of X minus one? You understand the meaning of zeros, right? Yeah. Zero is basically that number, which makes the entire expression become a zero. Yeah. What is the zero of X minus one? One. Correct. So make a one on the number line. Let's say I put it over here. What is the zero of X minus three? Three. Okay. So let's say one is here. Three is here. What is the zero of X minus X plus one? Minus one. Okay. So let me put it somewhere over here. What is the zero of X to the power six? Zero. Okay. So let me put it here. What is the zero of X minus two whole cube? X minus two whole cube. Pratik two. Yeah, it's two. So two will come somewhere over here. What is the zero of X plus two? Minus two. Correct. So let me put it over here. Fine. So as you can see on this number line, I have made the zeros of all these factors. So now this number line, because of these, you know, numbers that you have put on it, it has split the number line into intervals. So you have one interval here, one here, one here, one here, one here, one here. So seven intervals have been created by these numbers. Okay. All right. So this is clear to everybody. Make a number line, plot the zeros of that, you know, factor sitting and on that number line. Okay. This step is clear. Okay. Next step is very, very important. Next step is I'm not going to assign signs to these intervals. So what I'm going to do, I'm going to now assign signs to the intervals that have been created. Now, signs of what? Signs of this expression. And you must be thinking, how do I do that? So let us take an interval. Let's take the right most interval. Okay. Always start with the right most interval. Okay. Take any number more than three. Take any number more than three. Give me, give me one such number more than three. Okay. Four. Most of you are taking four. Okay. So if I put a four in this expression in place of X, the answer that comes out from there, what is the sign of that answer? So if I put a four in place of X, four minus one square positive. This is also positive, positive, positive, positive, positive. So what basically comes is the sign of this expression in this entire interval. Are you must be wondering, sir, you only took one value and then you generalize it for the whole interval. Yes, that is the beauty of it. Okay. So your entire function will be positive in this interval. Whether you take four, whether you take 4.5, whether you take 100, whether you take one million, whether you take one billion. If you put in this expression, you will always get a positive answer. If you don't believe me, you can try for other values also. Okay. Any value more than three if you take the sign of this expression, let's say I call this as Y, your Y will always be positive in this interval. Got it? Okay. Now most of you would be thinking, sir, are you going to do the same for other six intervals? Yes, but not in the same way as what I did for the right most. Now I'm going to tell you a rule. For assigning signs in the interval, we follow a rule. Start moving towards the left. Okay. Let's say if I start moving towards the left, the first number that I get is a three. Correct. Three comes from which factor? Three comes from this guy. Isn't it? Correct. What is the parity of the power over here? Parity means is it even or odd? Odd. Correct. The rule is if you have odd power on the factor from which that number is coming, then switch the sign. So this plus will become minus in this interval. So you don't have to put any value between two and three to check what will be the sign. The sign will be negative. Okay. Now obviously a question would be arising. Sir, what if it was even then what do you do? Then you retain the sign. Clear. So odd switch even retain. So I hope everybody has put down their pens and listening to me properly because if you make an error in understanding this, you will end up getting a lot of incorrect answers. Okay. So please listen to me very, very carefully writing and all I'll give you the time to do it. Not to worry. In fact, we'll take many questions also. Okay. So three second care. What is the next number that you see? Two. Two comes from which power? Two, two, two, two, two comes from this guy. So in this case, what is the parity of this even or odd? Odd. Odd means switch. So whatever you had here, that will switch and now it will become a plus. Got it. Clear. Okay. Keep moving to the left. Next number you see is a one. One comes from this guy. Correct. And this guy has got an even power. So even power means retain. So plus will remain a plus. Understood. Does it make sense? Does it make sense? Okay. Next number zero. Zero comes from this particular factor. And in this particular factor, you have power as even. So you will still retain the sign. So this plus will remain plus. Understood. Okay. Now you tell me what will be the sign coming here. Very good. Minus sign. Excellent guys. That means you're all listening to me very, very carefully. Okay. Because this minus one comes from this factor and this factor has got one as its power. So this is an odd number. Okay. So odd means switching of sign will happen. Okay. Tell me for this now. The last interval. Positive. Right. Because plus minus two has come from this factor and this has got odd power. So this will be plus. Is it fine? So this is the most critical step because if you make a mistake over here, then that problem will go for a toss. Now. What are we trying to solve here? We're trying to solve for which intervals. For what values of X is this entire expression on the left greater than or equal to zero. So greater than equal to zero means either it is positive or it is zero. Right. So let us see in which intervals are we getting the sign as positive. So I'm getting the sign as positive in minus infinity to minus two. I'm getting the sign positive in minus one to zero. I'm getting it in zero to one. I'm getting it in one to two. And I'm getting it in three to infinity. Right. Yes or no. Yes or no. Okay. So now what I'm going to do is I'm going to take a union of these intervals, but this interval is not complete till we put the brackets. Okay. So let's put the brackets around it. Remember minus infinity is always plus infinity is always around brackets. Okay. So never include infinity and minus infinity in your answer. Minus two. Can I include minus two? You'll say no sir. Don't even try to include minus two because if you include minus two as your X value, your entire expression will become undefined. Isn't it? So there will be a round bracket here. Right. Okay. So write a union here because you're going to take union of all these intervals. Similarly minus one. Can I include a minus one? You can say yes sir. You can include a minus one because at minus one, the expression will become zero. And that is a part and parcel of the inequality. Right. So I can include minus one. Fine. Is it clear? Why did I include minus one? Why did I put a square bracket next to it? Because I'll explain this again because at minus one, your entire function will become a zero and equal to zero is acceptable. No, I can have the answer as zero. Correct. What about zero? Should I put round brackets next to it? Or should I put square bracket next to it? Around. Correct. Okay. Because if I put X as zero, things will become undefined on the left. Sorry on the rational function. So this will also be round bracket because both are zero anyhow, and I have to put a union in between. What about one square bracket or round bracket? One. Can I include one in my answer? Yes. Square bracket. Correct. So put a union here. Two. Can I include it? Two. Can I include it? No. So what bracket should I put here? Round bracket again. Absolutely. Okay. Three. Can I include? Yes. So I'll put a square bracket. Okay. So your answer for X, that means your inequality that I give you here, the solution of this inequality will be this. Okay. So this is basically the answer to this question. Now the process is clear to everybody, right? Now what is the reason for this process? That is something which I'm going to explain you right now. Two factors have the same zero means you can club it. For example, if you had X minus two to the power three, and there's another X minus two to the power of four. So club it, write it as X minus two to the power seven. Why would you write it separately? Okay. Now I will explain you the reason behind this change of sign. Why do we change sign when the number that we are crossing is coming from a factor which has got odd power. And why do we retain the sign? If the number that we're trying to cross has come from a factor which has got even power. Let's try to understand this. See, why did we change sign along three? See, let's say X minus three to the power five. I write it separately and other terms. I'm clubbing it up as some p of X. Okay. Now understand if you are on the right side of three, let's say 3.0001. And if this was positive, right? Please note that whatever was the sign of this, that is not going to change. Right. If you move to the left side of three, see, let's say if it was, let's say if it was a positive number. Okay. And this was also a positive number. So this will remain positive. Even if you take X as 2.9999. But if this is positive and this becomes negative, overall, this sign will become negative. Right. So this entire expression will become negative in this zone. So if you're crossing a number, which is basically coming from a factor having odd power to the right and left of that number, there would be a change of the sign happening. But such a thing is not going to be seen if you have a factor subjected to even power. Let's say if I talk about X minus one. So let's say if I had X minus one square, let's say if it is positive right now, if you are at 1.001, even if you are at 0.999, this will still be positive because the power here is even. So when you're crossing that number, there will not be any change in the sign created because that particular factor was subjected to even power. Are you getting my one? Graphically speaking, the function will not change its sides. That means if it is above the X axis, it will remain above the X axis when the power or the number that basically you're talking about or the zeros that you're basically talking about is subjected to an even power factor. Okay. And it is going to go to the other side. It's going to go to the negative side. If at all it is going to cross a number which is coming from a factor subjected to odd power. Okay. See, it is not going to change because, I'll just take a simple example. Let's say for a particular number, if I take an example of maybe let's say one. Okay. Now, if you take a one, this entire guy, this entire set of numbers here. Okay. They will not. Let's say I put a one. Okay. Or let's say a value which is slightly more than one. What will be the sign of this number? Let me write it again because I don't want to make that diagram more dirty. Okay. So the other numbers and the other terms I'm writing in yellow. Yeah. See this. Okay. Let's say I call this expression. Hold this whole of this as P of X. So I'm just calling it as P of X. Now, if let's say you have, you have put a number 1.01. Okay. This will be what positive. This will be positive. And what will be the sign of this whole P of X? Tell me. If I put 1.01, this will be negative. This will be positive. This will be positive. This will be negative again. This will be positive. So overall, this will be positive, right? If I'm not mistaken. If I put X as 0.99 also, then see what will happen. This will still remain positive. This guy will still remain positive. And here what will happen? This will again be negative. I'm just writing it here. This will again be positive. This will again be positive. This will again be negative. This will again be positive. So overall, this still remains positive. Correct. So if you see the net sign was positive here also, then that sign is positive here also. So there is no change in the sign when you are crossing that number one. So whatever was the sign here, the same sign happened over here. So this is what I have incorporated as a rule so that you don't waste your time when you're solving it. Not clear, also live. Yeah, good question asked. Is it fine? Is this rule clear to everyone? Because I'm now going to give you a few questions and then I'm going to give you a break. No, no, no. I'm going to give you one question at least. I know you all are hungry. You want to take a break. Just one question I'll give you. Okay. Solve the inequality. Maybe I'll take a very simple question to begin with. Yes, yes. You can always take examples of a number in that interval and know the sign. But trust me, this method which I told you is much faster than that. But I'm leaving up to you to take a call. So please note the left hand side is a polynomial. But nevertheless, polynomial is also a rational function. After this question, we may take a break and when I return from the other side of the break, we will probably take more examples and then start with our next type of rational functions. Okay, Samarth. Okay, Satdeep. So Samarth answer is slightly different from Satdeep in terms of brackets. I hope all of you know the relevance of round brackets and square brackets. So request Satdeep to check your brackets. Okay, Sanjana. Okay, now Satdeep has changed it. Okay. Pramod has a completely different answer. Okay, Pramod. Rishabh also shows a completely different answer. Okay, Mahesh. Okay, Vasudev. Fine. So I think let's discuss. We have already spent a considerable amount of time on this. So first make a number line and make the zeros of this particular factor on the number line. So I can see a minus two. I can see a one and I can see a three. Okay. Now, only for the right interval, only for the right most interval, we need to pick up a sample value. So pick up anything more than three. Anything more than three. Let's say I take a four. Okay, so four will make this positive, this anyways will be positive. And this will also be positive. Okay. Now start moving to the left. So as I'm crossing, as I'm moving to the left, the first number I cross or the first zero which I cross is eight is three. And that zero comes from x minus three factor, which is subjected to odd power. So there will be a switch of sign. Right. Next comes from x minus one. So x minus one is having again an odd power. So there will be again a switch of sign. And then minus two comes from x plus two and which is subjected to even power. So there will be a retention of sign. Correct. No. Correct. No. Okay. So if I, if I have to say which interval is it less than zero, less than zero means negative. So I've only written negative in the interval one, two, three. There's all the interval where I've written a negative sign. And please note, since you cannot have this expression equal to zero, one cannot be included. And you also cannot include three because that will make it zero. So your answer to this question is open interval one, two, three. Okay. So I think the, let me see who's the first one to get the site. Samarth Rao got the site. Well done, Samarth. So with this, we'll take a small break and on the other side of the break, we will take few more problems on solving wavy curve, science came questions and then we'll jump to the type three problems. So right now is three or five. Let's see each other at three, 20 p.m. sharp. Thank you. Enjoy your break. So let's take another question. Let's take the previous slide one. Previous slide, you want me to go? Okay, sure. I'll go, I'll go to the previous slide. Yeah, tell me. All fine. Guys and girls earnest request not to keep any doubts with respect to this concept because this is going to be a deciding factor in many of our subsequent concepts. Okay. So please make sure you are 110% sure about how this rule works. Okay. May I go to the question now? Yes. Okay. Thanks a lot, Mayesh. So I have another question. So let's take this x plus four to the power of 100 x minus two to the power of 1001 x plus one to the power of let's say one x minus one to the power of let's say two all divided by x to the power of 600 x plus two to the power of 501. This is less than equal to zero. Solve this in equation. Solve this in equation. This time I don't want anybody to make any mistake. So read this question carefully. Let's not assume the powers. I thought the answer would be up by now. So I'm not solving it. Meanwhile, you are solving. I'm going to just make my zeroes here. Zeroes of these factors. So minus four, I can see, I can see a minus two. I can see a minus one. I can see a zero. I can see a one and I can see a two. Or maybe it's taking time for you to type it out. Okay. Now I understood why it is taking so much time to answer. Okay. Okay. Okay. So two people have answered so far. I'm waiting for this. Okay. Sanjana, it's very long. You don't want to type it. It's fine. It's fine. Absolutely fine. I'm not forcing anybody to type out an answer which you feel is too, too lengthy to type. Okay. It's fine. Oh my God. Again, you read the question. No, this time, Vasudev. Let's discuss this out. So if I take the right most interval, I can take any dummy value, maybe more than two, let's say three. Everything is going to be positive. Right. Now here, many people ask me this question. Sir, by default, is it positive always? I would say no. Okay. So don't take chances. There are certain situations when you realize that the right most interval can also become negative. So this is not a rule that you should be making for yourself. Better to invest that 15, 20 second of your time to figure out whether you should be positive or negative because everything is linked to that. So that is our pointer. Right. So our change of sign or our retention of sign is keeping this right most interval sign into our mind. Right. So if that becomes wrong, my dear, your answer will become Ulta of whatever it was supposed to be. Okay. So please be very, very careful. So let's not make a rule that it is always positive. Okay. Hardly takes 10 seconds to figure out whether it should be positive or not. Okay. Now let's start moving to the left. So the first number that we see is two. Two is coming from x minus two, which has got an odd power. Odd means odd means switch. So plus will become minus. Yes or no? Yes or no? Next, one. One has come from x minus one. Right. x minus one has got even power. So there will be a retention of sign. So minus will remain minus. Then zero. Zero comes from x to the past 600. 600 is even power. Even means again the tension. So minus will remain minus. Correct. Minus one comes from x plus one. x plus one has got odd power. So if nothing is there, means power is one. So in that case, there will be a switching of sign. Okay. Next, minus two. Minus two has come from x plus two, which has got again an odd power. Odd power means again switching of sign. Correct. Then minus four. Minus four comes from x plus four and has got even power. That means again a retention of sign. So please check. This should have been the sign that you should have got. They should have got. Okay. All right. So let's now write down those intervals where my function is either negative. So this means either negative or zero. Okay. So let's write down the interval. So it's minus infinity to minus four, minus four to minus two, minus one to zero, zero to one, one to two. Okay. Now let's write down the brackets. Round brackets minus four. Minus four comes from a factor on the top. So this is included. Okay. So union minus four included minus two. Minus two comes from factor in the denominator. So this cannot be included. Similarly, minus one comes from a factor on the top, which is included. Zero comes from a factor on the bottom. So which cannot be included. Again, one comes from a factor on the top, which is included. Two comes from a factor in the top, which is again included. Is that fine? Any questions? Any concerns with this so far? Okay. Now please note that you should have actually included minus four to minus one and one in your interval. So is minus four included? Yes. Minus one included? Yes. One included? Yes. Two included? Yes. So that's all you need to write. So this becomes your answer to the question. Now, many people try to coalesce these intervals. Coalesce means try to make them one. Like you can also do a single, you know, in a single statement you can write minus infinity to minus two directly. You don't have to write, you don't have to split it unnecessarily at minus four. Yes or no? Similarly, similarly, you can coalesce this interval. So you can directly go from zero to two also without breaking it up at one. So instead of writing such a long answer, you can cut short your answer in this way also. Getting my point. Now, many people say, can I also write the answer like this? Minus infinity to minus two union minus one to two directly and they will just exclude a zero from it because zero is coming in between. So even this is acceptable. So all of these are acceptable answers to the same question. Is it fine? Any questions? Any concerns? Okay, good. So with this, I think we have we are now heading towards the third type of question. So type three. Okay. So here I will take a question where there is a rational function which has got a linear or a constant on the numerator and has got a quadratic in the denominator. Okay. So it has got a linear polynomial or a constant polynomial on the numerator and has got a quadratic polynomial in the denominator. Okay. Let's take once a simple example, which is actually picked up from your NCRK textbook, this expression. Okay. So I've taken a constant and I've taken a quadratic. Question is, let's find out its domain and range. Now, would you like to try it first or should I explain you how to do it? Tell me, would you like to try it out? You want to try it out? Okay. Go ahead. Try it out. I'm giving you, I'm giving you time to try it out. Very good, Sathdeep. Domain is simplest. Domain, I don't think so. Domain is bothering anybody over here. But what people are bothered is by the range. Correct, Sanjana. Sanjana and Sathdeep. So far, both of you are right in your domain at least. Vikram, first try it out. Okay. First put in your efforts, then we'll explain you the, then I'll explain the solution for this. Pratik, are you sure you are covering everything or are you missing out on something? Okay. Very good. I'm getting answers for the domain for most of you. I'm still waiting for range. Okay. Anybody who is coming with the range answer? Nobody? Okay. Luxure, that's your range or that's your domain? Range. Okay. Fine. Let's discuss it out, my dears. Domain is plain and simple. Domain is going to be the same formula that we have been using since our, you know, the beginning of our rational functions. R minus those values of X, which will make the denominator become a zero. Okay. First of all, this becoming zero means X square is one. That means X is plus minus one. So your R minus one and minus one. So this will become your domain of the given function. Right. Nobody has any issue with the domain. Pratik. Okay. Is it fine? Now what about range? Range the idea is more or less the same. We will call this as Y. Okay. And we'll try to make X the subject of the formula. How do I do that? Let's check. So first of all cross multiply, I mean multiply the denominator to the left side. So this will give you X square Y is Y minus one. So X square is Y minus one by Y. Okay. Now please note you don't have to exactly make X the subject of the formula. Even X square will suffice for me. Now listen to this very, very carefully, my dear. Very, very important. If your X belongs to all real numbers except one and minus one, then what can you say about X square? In which interval will it belong? No. Write it on the chat box instead of all real numbers. No. All positives. No. The square of a number which doesn't include one and minus one can mean what interval? Our simple zero to infinity except one. No. Why zero to infinity? Because you're squaring a real number. If you're squaring a real number, your answer should be greater than equal to zero. But you can't be having a one because your X is not one or minus one. Whose square happens to be one. So you can't get one from your squaring. Now what does it mean? It means your Y minus one by Y is greater than equal to zero and cannot be one. This guy cannot be one. But this is always true because this anyways cannot be one. True always. Right? Because if you put this equal to one, what will happen? Let's check. If you put this equal to one, you'll end up getting a shock of your life. So this will anyways not be one, but this should at least be positive. Correct? No. And here I can use my wavey curve because this is a rational function on the left. It's just that the function is now in terms of Y and the examples which I was taking was in terms of X. So name of the variable doesn't change the concept. So if you make the zeros, these are the zeros, zeros of this guy is one, zero of this guy is zero. Correct? Put the signs. So for putting the sign, I will put the sign of the rightmost interval first. So for that, I'm choosing some, you know, test value as two. So two makes it positive. And since all of them have odd powers, you can start switching the sign minus and this will become plus. Okay? Now, since you want it to be greater than equal to zero, you have to always choose positive. Correct? So your answer will be minus infinity to zero union one to infinity. Now mind you, zero cannot be included, but one can be included. So this becomes your range of the function. So this is your answer for the range. And I don't think so. Anybody got this right? I don't think so. Anybody got this right? But why you want to take cases when there's a wavey curve? Why to take cases? Wavey curve is like, you know, making my life easy. Let me use it. Is it fine? Any questions? Any questions with this approach? So please note, the process of finding the range is not as easy as finding the domain. So you need to practice more range based questions. In fact, both domain and range based questions. Anything that you would like me to revisit here or explain once again, kindly message me or speak out. I'll be more than happy to help you out again. Sir, the inequality is good. Yes, from there. See, your xx square expression is this right? And since your x belongs to real numbers, except one and minus one, your x square will belong to this interval. Is this agreed upon by you? Yes. So x square is what x square is this guy only. So this guy should be greater than equal to zero. This is what greater than equal to zero. And at the same time, it should not be one. And why can never be one? In this case, this expression can never be one because if you equate it to one, you'll end up getting that something like this is happening. So this will always be true. Another way of looking at it is another way of looking at it is this will never be one for all real numbers. I was asking after that. Yes, let me complete. And this will be positive only for this interval. So let me not write range here. Let me write like this. Okay, now and means you have to take the overlap of these two situations. So what I'm going to do is I'm showing another perspective of it slightly more, I can say detailed one. So you are taking an intersection of these two intervals. So this interval says this interval says I have to be on the left of zero or more than one. And this interval says I can be any real number. What is the overlap of the two? Where do you think the two lines are overlapping here and here? Correct. So overlap of these two will ultimately lead to your range, which is the same as what you wrote here. So this becomes your range of the function. Is it fine? Yes, sir. Got it? Okay. Sir, I did without taking that x square and all. Ah, see, whatever approach you take. So what, how did you do it without taking x square? I didn't take x square. So root of y minus one by y, then the same then wave occurred, but it took lesser than equal to and that's all. Right? So if you take lesser than equal to, then you get like the values for which it's negative, then you can just do real numbers minus that, then you get the same answer. One second, one second, one second. You, you wrote x square as y minus one by y, then what did you do next? Root. Root. So it will be plus minus this. Up below, next one. Then for y minus one by y, wave occurred. For y minus one by y, yes? Yeah, because less than equal to zero. Less than equal to zero, absolutely wrong. So no, we can do that. Then in the last, you get the values which is, so I got the same answer. No, sir. No, sir. This quantity can never be negative. Else your x will become non real. Exactly. That's why you exclude those values. The end will exclude R minus, real numbers minus. Say that, no, you excluded those values. Okay. Then how did you ensure that it is not equal to, it is not equal to one or minus one? So that didn't do. Correct. This is the thing, watch what people are doing nowadays. They're just, without the domain into consideration, they're trying to solve the problem. And this is a simple problem. You may get the right answer in that way also, but when it becomes, question becomes complicated and you're not following an exhaustive process, there are chances that your domain, you will miss out something, range you will miss out something. Okay. So I guess we do in that method, you can't check it. Sorry? When you like take lesser and solve it, I don't think you can check. If you take lesser than and do it, then you don't think so? You can check it, like put it equal to and see the idea is this expression that you're talking about under root of plus minus under root of y minus one by y. This should be all real numbers except one and minus one. Correct. Now, if you have to make it all real numbers, this entire expression will boil down to you ensuring two things. Number one, this guy should be greater than equal to zero, which anyways has been addressed over here. Correct. And at the same time, y minus y should not be one because when only this is one under root of plus minus one will become plus minus one. So ultimately you have boiled down to the same equation, Mahesh. What's the point? Okay. So that approach provided it is taken in this way will also lead to the same answer. Let's take another question. Let's take another question. Find domain and the range for x by one plus x squared. Try this out. Okay. Domain everybody is getting right. Absolutely. Range is the one which is more important. Range is all real numbers. Okay. So no, not range. The first one was domain and after that is not for you, Mahesh. You have you have perfectly answered it. I'm talking from Vasudev's answer. Okay. Should we discuss it out or should we wait for more time? Okay, got them. Wait, wait. Okay. Fine. I'm waiting. See you solving the question gives me more pleasure, right? Because ultimately you are learning here. You can find a lot of these questions in Aadi Sharma, MCQ part. So last, last, you can say a few pages of the chapter you will see they have given some MCQs. So maybe I would like you to suggest trying those questions also. All right. So let's discuss it because we don't have much time. So here the domain will be all real numbers. Of course, excluding such access for which this will become zero, right? But trust me, this is not going to become zero ever for any real numbers. So this will be a null set. So it's like real number minus null set. So that's equal to a null set. Okay. Next, what will be the range for range? First of all, I'm going to take this as a Y and now I'm going to make X the subject of the formula. So while doing that, you would realize that you will end up getting a quadratic in X. So here making X the subject of the formula is not as simple as what it was in the previous few questions. Okay. Here X is coming like a quadratic, right? And this Y is acting like the A of the quadratic, then minus one is acting like a B and Y is acting like a C, right? So we will use, we will use the quadratic equation formula that is minus B plus minus B square minus four AC, okay? By two A, by two A as your X value, correct? Yes or no? So far so good. Anybody have, anybody has any issue till this step? Anyone has any issue till this step? Okay. Now let us recall, let us recall that we are dealing with real numbers. So this quantity that is one plus minus under root of one minus four Y square by two Y should be real numbers, isn't it? Because your X is real, isn't it? Because your X is real numbers, right? So this should be real. So when it is real, what are the two things that you can comment looking at the fact that this should be real? One thing for sure you can comment is that this term one minus four Y square which is subjected to under roots that should be greater than equal to zero. Because please note that we can never have a negative quantity subjected to an under root sign. Okay, so that quantity one minus four Y square should be greater than equal to zero. Everybody agrees with me on that? Everybody's fine with me on that? Okay. And second thing that possibly people will say is that Y should not be zero. Okay, but let me tell you this is subject to verification. So we'll verify it. Okay. So these are the two things that come in your mind when you're trying to say that this quantity should be real in nature, isn't it? So far so good. So these two conditions one and two must together be satisfied, isn't it? Any other restriction that you want to put on Y other than these two? No restriction. Okay, cool. Now let's check it out. So if you take this fellow one minus four Y square greater than equal to zero, so I'm now factorizing it like this, right? Can I apply my wavy curve on this? Can I apply my wavy curve on this? Yes. No, sir. I don't think so. Why not? These are factors. The factors here are having zeros as half and minus half. Why can't we apply wavy curve? Sir, sorry, sir, I got a little confused. Okay, we can, right? Now let us assign the signs to these intervals. What sign will come in the right most interval? Tell me plus or minus, plus or minus, plus or minus, write down on the chat box. Take any number more than half. Let's say take a one and put one everywhere in place of Y. What do you get? Negative sign, right? And since all these factors are having odd powers, there will be a switching of sign like this. Correct? Everybody's happy with the sign scheme. Okay. Now greater than equal to zero means positive or zero. So positive or zero will only be in this interval minus half to half. And since it is allowed to take zero, you will have a close interval. So from the first answer, you end up getting this. So this is our result of the first one. Now second one, and I can see Rishabh also saying that Y cannot be zero. Now please note that when you're saying Y is equal to X plus one by X square, if X is zero, Y can be zero. No, Rishabh. Just because Y came in the denominator here, I understood your concern that you're saying Y should not be zero. But look at the original expression. When X is zero, Y can be zero. No, this is your Y. Sir, but then the problem is that the first expression contradicts the second expression. So you don't know which one is correct. Which quantity is the first expression? Sir, the first expression says that Y can be zero. Second, when you express the equation in terms of Y, then you'll get zero. Even this doesn't contradict Vasudev. If you put Y as zero, you'll end up getting an indeterminate form which is zero by zero. This is considered to be an indeterminate form in mathematics. Sir, what's the difference between this and undefined? I'll tell you. Sometimes what happens when the quantities in the numerator and denominator both are zero zero each, your answers can be a finite value. Those are called indeterminate quantities. But there are many situations where if you have something divided by zero, you don't get a defined answer. That is an undefined quantity. So in such cases, you have to take a call on the basis whether can my original equation, can I get a zero from this by choosing any real X? The answer to that is yes. So this condition is actually an indeterminate form. It is not an undefined expression like how it used to happen in the previous cases. So if your numerator becomes a zero, then there is a chance that your expression may become indeterminate. Indeterminate means it can take a finite value. Means it need not be undefined expression. We will talk more about it in the limits chapter. So here the verdict is, this is ruled out. So your answer to the range just becomes minus half to half. Getting my point. You don't have to exclude a zero here, Rishabh. Sir, what about the second case in that? What second case? Sir, there are two cases of 1 minus 2y and 1 plus 2y greater than or equal to zero. Either both are greater than or equal to zero or both are less than. If you didn't get the entire agenda. I am taking care of everything simultaneously. Why are you taking cases? You want to do it by cases, your problem will take 3 to 4 minutes more. What is the whole point of learning wavey curve? You don't have to take cases. Both are positive or both are negative. That all you need not do. You may have learned that in school. But when it comes to solving an equality using your wavey curve, every condition is taken accounted for here. Yes, sir, got it. Wavey curve, why did we learn at all wavey curves? So that we don't have to make those cases. If I had to solve the case, why would I do all this wavey curve discussion at all? I would have done that only. So wavey curve circumvents that longer path and gets to the answer faster. Got the one? Got it. Sad. It's okay. I mean, what making a mistake should not make you sad. It should make you happy. Oh, I learned this. I was always making a mistake in this kind of question. Sir, I have doubts that I tried to do it in a different method. Yes, tell me your method. I will justify your method also. Tell me. Sir, I didn't get the same answer. I want to know what I did wrong in it. Tell me. Sir, so x divided by 1 plus x square. So the denominator is always positive for any real number. x divided by 1 plus x square. Sir, denominator is always positive. Yes. Sir, numerator could be positive or negative. Right. Sir, so this whole expression could be negative, zero or positive. In case the numerator is zero, then it will be zero. So it means everything it can take. Yes, sir. Set of all real numbers. Who told you? Just because they are positive and negative, it means it covers the entire real number line. Why? Look at the ratio. See, if I write one quantity by another quantity, right? And this can take the same value you are putting over here and this Q is never allowing P to be more than it. That means this guy is always more. Then this quantity will always be less than 1. Did you account for that? Oh, no, sir. I didn't account for that. So it is not as simple as saying that above one is positive, below one is negative and hence the ratio will be everything. No. Okay. Got it, sir. Got it. That's the difference between school teacher and this session, right? Okay. Now excluding zero is sometimes present in the options also and many people wrongly mark that option as well. Okay. So please be careful. Please be careful. Don't do those mistakes. Okay. Let's take another question. This time, in fact, another type I will take. This time I'll take a type where you have a quadratic by quadratic. In fact, this is the last type that we are going to, you know, take up quadratic polynomial by another quadratic polynomial. See, in your school, you should have asked your teacher, no Q. Why are you excluding zero? Right? Why are you excluding zero? Just because you had a zero in the denominator, but your expression will take zero when X is zero. These are the questions that you should be asking. I don't expect things to be straight away. Okay. Mam is saying so, sir is saying so, so it should be correct. Then that is not how science and mathematics is, is, you know, studied. You have to ask right sets, right set of questions. Okay. Okay. So here also I'll take one simple example. Let's say I take this function X square minus X plus one upon X square plus X plus one. In fact, I will show you the graph as well. Our graph is a very good way to learn the domain and the range. So our previous function was this, right? Y equal to X divided by X square plus one. Right? If you see from the graph, domain is everything because it goes all the way in both directions and range is between half to half. So 0.5 is the top edge and minus 0.5 is the bottom. Okay. So this is your, this is your range of the graph. Okay. And zero is very much taken. See the graph is passing through origin. Right? Okay. Anyways, let's take this question. We have to find the domain and the range for this particular function. Sir. Yes, sir. I don't think it's as the denominator is not rational, sir. You'll get X is equal to minus one plus or minus minus two, three. So, sir, but similar question like this in center module. That's fine. But what is the question? Sir, my question was like, I was solving the center module. There was something called EXP in the exercise of domain and range. What is EXP? EXP means e to the bar, exponent. Who had this concern that, you know, the X here is, what do you mean by solving for X? And why are you solving? But you'll have to find if it should not be equal to zero. Right? So if it is not equal to zero, say everything is accepted. What is the, what is wrong in that? Oh, yeah, sir. Okay. So, Pramod has given one response. Very good, Pramod. See, I'm more interested in knowing the range. That is what is, you know, the difficult of the two actually domain. I don't think so. Any issue, I'll write it down for the domain. So domain is going to be all real numbers except those values of X for which this guy will become a zero. Right? But you know that this fellow can never be zero. Why? This will never happen. That means it's a null set. Why it will never happen? It is because it is because X square plus X, X square plus X plus one, if you complete this square, it will look like this. Okay. So this quantity will always be more than three by four. So this can never be a zero. Because it is always greater than equal to three by four. So this will not give you any answer. So you don't exclude anything from real numbers and hence your domain is all real numbers. So getting domain is not at all difficult. It's super, super easy. Range is what I'm interested in. Okay. So Vasudev has given one answer for range. Okay. We'll see Vasudev. Anybody else? Okay, Mahesh. Mahesh has got completely different answers than Vasudev. Okay. Should we move on with the discussion for range? Okay. So, oh, you want some time? Fine. Fine. I'll wait. I'll wait. I'll wait. See, if you solve it, I'm more than happy. Okay, Vasudev, you want to change your answer? Fine. Acknowledged. Shall we now? Okay. The first thing that I would do here is again, try to write X in terms of Y. Right. So for that, I'll just take the denominator to the left side. This is what I'll end up getting. Okay. Let's bring things to one side. Okay. So I happily see a quadratic sitting over here. Okay. Very good, Samath. Pramath, very good. Okay. So what do you see here is a quadratic in X. Right. So what we see is a quadratic in X. So let's use our Shridhar Acharya formula to solve this question. So when we use Shridhar Acharya formula, you get minus B. Minus B will be a negative of Y plus 1 because B is your Y plus 1 plus minus B square minus 4 AC, minus 4 AC. Okay. Divide it by 2A. Is this fine? Rishabh, very good Rishabh. Okay. Now, if you want this to be real, I'm writing it as big. If you want it to be real, what are the things that you need to keep into your mind? So the first thing you'll say, sir, I don't want this under root thingy to become to be a negative number because then under root of a negative number will give me non-realances. Right. So the first thing that we need to acknowledge and keep in our mind is that this guy should be greater than equal to 0. Of course, I will simplify it a little later on, but as of now, I'm just pinning down the possibilities for this quantity to become a real. Second thing, which you possibly will say that Y should not be 1 because this guy should not be 0. But again, this is subject to verification. We'll verify it a little later on. Okay. Now, let's start with the first one. Let's start with the first one. So the first guy says this fellow, this I can write it as something like this, minus twice Y minus 1 the whole square. Why I'm writing like this? Because I can use X square minus Y square formula or a square minus B square formula. So it'll be A plus B into A minus B. Correct. Please highlight if I missed out anything. So this will be 3Y minus 1 and this is going to be, if I'm not mistaken, 3 minus Y. Correct. So again, it's a rational function inequality. So may I call upon wavy curves to help me for this? Right. So wherever there's a rational function inequality, think no further than using your wavy curve. Right. So wavy curve is one shot solution. You don't have to make cases unnecessarily. You may have learned how to, you know, solve this by making cases, but that is going to take double the time to do the same work. Okay. You may be, you may like to use that in your school exams, but wavy curve is single shot, one shot, one stop solution for all these rational function inequalities provided the structure is the same as what we had discussed while discussing the concept. So here, what will be the zeros? One third and three. Okay. Okay. Now take the value, which is more than three, let's say I take a four, everything will be a negative. So this will be negative. Now see this is one occasion where you started with negative on the right most. So never, never take it positive by default. And since all these factors are coming from odd power, there will be a switching of sign. Yes or no? So greater than equal to zero, the interval that you are getting is one by three to three. And since it is greater than equal, equal is there, you can include both of them. Correct. Now the second possibility says why should not be one and I have to be careful about it because one comes somewhere in between. So should I actually remove one or should I not remove one? That is something which my question itself will tell me. So I don't think you should remove one. Okay. Let's see why. So if you equate it to one, do I get a real X for this? You will say yes. If you just equate it, you will end up getting X value as zero. So if I put X as zero, I will get Y as one. So there is no reason for you to remove Y equal to one. Okay. So this itself, this will be ruled out. You will not pay attention to this. Okay. Because it has failed the verification. So the only answer that you are going to acknowledge is this fellow. So this becomes your a range of the function. Is it fine? Any questions? Any concerns? Sir, I had a doubt. Yes, sir. Tell me, sir. Sir, I think you might be able to do this without getting, without changing the equation in terms of the Y and solving the quadratic equation. Okay. Tell me the process. Sir, I think you might be able to do it directly from here. You might be able to judge like that X square minus X plus one divided by X square plus X plus one is always lesser than one or something like that. I've been trying to do that, but I'm not getting the correct answer. Could you please show how to do it from that? Not the way to do it. Because that is not the way to do it. How can you say that this is always lesser than this? There's no such, X is a variable. X could be any real number. So how do you say that this quantity is always going to be more than this quantity just because there's a minus sitting out there? Because we have got our range to be three also. That means this guy can be three times this also. Yes, sir. Yes, sir. There is no such way to say by looking at the expression, you see, we are not astrologers over here. We are mathematicians. At least we are acting like mathematicians. So we have to be logical in the way we are writing steps. Yes, sir. Got it. Sir, we can't use baby curve in school exam. Why not? Is there any instruction given by the school teachers that you can't use baby curve? I mean, Mahesh is saying so. Is it true? Sir, they only used baby curve, sir. Sir, no, sir. I was asking. There was a question asking if we can use. I mean, see the best person to answer this is your school teachers themselves. But if she's already using it or he's already using it, then you can use it. Why not? Nobody's stopping you. Okay, let's take another question. X square minus one by X square plus one. For this function, find the domain of the function and the range of the function. Yeah, domain is very easy and everybody will get the domain. Absolutely right. Got them correct. Some are correct. Domain wise, everybody is fine. No issues with domain. Range is where the story gets stuck. But I'm sure now that you know the process, nobody should be getting stuck now. Okay, Vasudev, very good. So Vasudev has come with Vasudev, Pramod, very good. Excellent. Come on. My very good doesn't mean right. Okay, I keep saying very good because I like people answering. Whether they are right or wrong, that is immaterial to me. I always see the efforts. Okay, just put in the effort, put in the right effort. Things will start getting right in some time. Okay, but I want people to participate. I can see many of you are not participating. First of all, I don't know for some reason the attendance is fluctuating between 20 and 21, 20 and 21. There's one person who is continuously leaving and joining back and again, back and forth. I don't know who is that person. Maybe I'll come to know from the Zoom report. Anybody who's getting constantly disconnected, I'll give one more minute for people who are trying hard for range. Please wrap this up in one more minute. Our next class, I will wind up this chapter because I think we need to now start with trigonometry. That's a long pending concept. So through this chapter, I have already tried to cover functions, sets, relations, functions and inequalities. So all your needs for set relation functions and inequalities, that means four chapters have already been taken into consideration under this single chapter. Okay, our next class, that is the last, that should be the last class for us on functions. I will be covering up irrational functions. I'll be covering up exponential logarithmic and special functions. So these four types of functions will be taken up in the next session. Okay. Oh my God, Sadeep has a different answer. Sanjana has a different one. Reshev has a different one. Oh my God, this is something which I don't like. People giving different different answers for the same thing. Okay, anyways, let's check. So first of all, domain, what's the domain? So domain will be all real numbers except those X for which the denominator would become a zero. But you know for sure that there is no real X for which X square plus one will become a zero. So you'll end up getting excluding nothing from R. So your answer is actually R. Now, what about the range? Let's talk about it. So for range, what I'm going to do is the same approach, I'm going to write X in terms of Y, X in terms of Y. So when I cross multiply, when I take the denominator to the other side, this is what I'm going to see. Okay. So Y plus one is equal to X square minus X square Y. So you end up getting X square one minus Y. Okay. So X square will be equal to Y plus one by one minus Y. Is it fine? So far so good. Any issues with this making X square the subject of the formula? Is it fine? Now, you can do it without making a quadratic also because clearly you can get directly your X square. Now see, since X belongs to real numbers, can I say X square will belong to? Complete it. If X belongs to real number, X square will belong to real numbers minus those which are not perfect squares. What are you saying? Zero to infinity, right? Sir, because then if X is like real number, then X square will be 149 like that. Oh, if for you, natural numbers are only real numbers. Oh, sorry, sorry. I forgot about the others. Hey, wow, wow, Asudeh, didn't sleep properly last night. Okay, anyways, so coming back. So since X square is this guy, can I say Y plus one by one minus Y is greater than equal to zero. Now, the moment I see this, what comes in my mind? Way we curve because it's a rational function inequality, correct? Yes or no? Let's put the signs. So what will be the sign in the right most interval? Positive or negative? Sir, I guess it's minus. And you guess it, guess is absolutely right, okay? So if you put it to here, everything will become negative, right? So three by negative one is a negative number. And now all these factors have odd powers. So it keeps switching from one sign to the other, right? Now you're trying to solve when it is greater than equal to zero greater than equal to zero means either zero or positive. So minus one to one. Now please note, minus one can be included, but one cannot be included. Are you getting this point, right? So your Y will be in this interval, which means your range of the function is going to be this. So this is your answer to this question. Is it fine? Any questions? Any questions, any concerns? There is one by two order even. One by two. So because I took the root and it, so if I have to apply that, if one is a number is odd, power is odd, so I can have the minus half there. One by two is order even means I didn't get your question. So sir, you took x square, which x square, I mean, I took x is for the root of the thing. So for me, it would be Y plus one to the power half. Okay, you wrote it something like this, then go ahead. What did you do then? So Y plus one to the power half and one by one minus five to the power half. Achha, one more important thing. Is this the same as, let's say I write it like this under root A, B. Is it same as saying under root A by under root B? I guess that's what. Okay, please note, in this case, the only restriction that you have is A by B should be positive. Correct? Or non-negative, so as to say. But here you have a restriction that even A should be this and even your B should be this. Right? So is there any difference between this and this? Here the ratio is positive or here the ratio is non-negative, but here the individually they should be positive or at least A should be non-negative. Right? So when you write it as under root of Y plus one by under root of Y minus one, you have already put extra restrictions on Y which is meant to be not good for your problem solving. You're going to miss out on a lot of values of Y. Oh yes, okay. That's a mistake which many people do actually, hence I thought I would discuss it with you. Now someone is asking why it is not under the, under the, yeah, so the verification step is already taken care of. Right? See, if you would, if you want to write it as a quadratic, let's say here itself, I would write it like it as a quadratic. So X square one minus Y minus Y plus one equal to zero. So when you solve it as a quadratic, X will be minus B which is zero plus minus under root of B square minus four AC. So minus four AC will be four Y minus one and sorry, one minus Y and Y plus one by two A. Correct? So there are two things that you need to ensure. One is this guy should be, this guy should be greater than equal to zero. And second step is Y should not be one. Okay? So you are basically trying to approach from this angle. Right? So here you have to verify this part that is what you are trying to say. Correct? Summared? Okay. Now this is as good as the wavy curves. So minus one one, please note the sign here. This will be plus, sorry, this will be minus plus minus. So you end up getting Y value as minus one to one closed. Correct? No? Yes, sir. That's what I thought. Now see, here Y genuinely cannot be one. This is also true. The reason being, if I put Y as one, see what will happen. I'm putting one as Y. So if you do this, sorry, the sign was minus on top. Right? Yeah. You will end up getting something like this. X square minus one is equal to X square plus one, which means you're trying to say minus one is equal to one. So this is, this is generally to be taken into consideration. You know, why will it not be one? Because if Y is one, this non-mathematical result will come out, which is definitely not acceptable. So hence, this one which you had included here has to be excluded finally. So my method automatically takes that into consideration. So if you take this approach, that is your second approach, even that will give you the same result. Is that fine, Samarth? Make sense? Yes, sir. Okay. So dear all, I'll stop over here because the next concept that we are going to start with is your rational function. I don't think so. We have, you know, any time left, hardly one minute will be left. So next class, I'll just tell you the agenda. Next class, I should be able to finish off. Number one, domain and range of irrational function, domain and range of irrational function. This is very, very important. We'll talk about domain and range of exponential and logarithmic function, exponential and logarithmic function. Okay, I will not be talking about domain range of trigonometric function because anyways, that's a agenda that I would be taking up under trigonometry. And third, we'll take up modulus function. Let me write like this, mod function. One one example, I will not go into details of it. One one example of mod gif, lif, okay, fraction part, okay, max min, signum. Okay, so one one example will take up on this, this concept. Okay. So I think next class will be second Saturday, right? Second Saturday is offline, no? Yeah, sir. Okay, so we'll take this in the class itself. ESS, in the monthly test, till ever, wherever we complete, that will be included in the monthly test. Okay. Is it fine?