 So in today's session, we are going to resume with the concept of domain and range for polynomial functions last class. We covered trigonometry. Sorry, quadratic polynomials. Isn't it? Okay, so today I'm going to start with cubic polynomial, cubic polynomial. Cubic polynomial means degree three polynomial. Okay, now a typical cubic polynomial is going to look like a x cube plus b x square plus cx plus d. This is a cubic polynomial. Remember in the bridge course, we didn't go beyond or you can say pure cubic polynomial. We didn't have anything other than x cube. Right. But in reality, when you talk about cubic polynomial, you may have terms containing x square x and constants. Okay. However, our main agenda is not to go into the depth of sketching the graph, but to know the domain and the range for cubic polynomial because that is what is going to be coming up for you in your school exam also. And of course, J and other competitive exams as well. So any cubic polynomial, if you see it has a very interesting trend depending upon the leading coefficient. So the leading coefficient plays a very, very vital role here. If your leading coefficient is positive, if your leading coefficient is positive, please note the following trend. The trend is, as x will become very, very large, the function f of x will also become very, very large. Okay, you want to call f of x, y, whatever you want to call it. Okay, so when your leading coefficient is positive, your x, if it becomes very, very large, your y value will also become very, very large. Okay. And if your x becomes negatively very, very large, then your f of x will also become negatively very, very large. Okay. Why? Because see, it's very simple. If this is positive, and if your x is very high, it doesn't matter whatever is the sign of BCND, because leading coefficient or this term x cube will have more dominance as compared to the other terms. So if you keep your x very large, and your a is positive, then your y, which is your f of x, let me write it down here, your y, that will also become very, very large. Okay. And if your x is negatively very, very large, it doesn't matter whatever is your BCD, whether they are positive, negative, zero, because of this dominance of this term, your y will also become negatively very, very large. Is it fine? Now under such a situation, when you sketch the graph of this type of function, you would realize that this graph will always end with an arm up and an arm down. See the motion of the arrows over here. So it will be something like this. I hope you can see me on my camera, on the camera. So one hand will be up, and one hand will be down for the graph. So it is trying to indicate, what is it trying to indicate? It tries to indicate that as your x is infinitely big, so will be your y. And as x is infinitely small, or you can say negative infinity, so will be your y as well. Are you getting my point? So one such typical example that you know of is your x-cube graph. So one example I will give you x-cube graph, since you have done this in your bridge course, I would like to revisit this graph for you again. In x-cube graph, what do you see? Let me make my axis here. In x-cube graph, the graph is like this. The graph is like this. Yeah. So just focus on these two arms. Just focus on these two arms. Where is it heading towards? It's heading towards plus infinity as your x goes to plus infinity. And it's heading towards minus infinity as your x goes to minus infinity. Okay. Now in this case, you would see that there is only one point that we talked about in the bridge course, and that was this point, which is called the inflection point. Okay. What is inflection point? You need not bother right now. It's too early to talk about inflection point. Inflection point is anyways our subject matter when we are doing calculus. Okay. So here what is important for us to know is that a cubic polynomial graph can have several peaks and valleys. Okay. When I say several, it cannot exceed more than one peak and one valley. Okay. So I have not drawn it in many of the situations, but today I will try to draw it. Let me go to the GeoGebra screen over here. I'll try to show you from there. Okay. So let's draw a typical cubic polynomial graph having some values of BC and D and of course your A. So let's say I draw y is equal to x cube minus let's say 4x square. Okay. Plus let's say 2x plus 2. Okay. If you see this graph, again, it is following the trend which I told you a little while ago, right? Its top arm here is going towards plus infinity as your x is going towards plus infinity. And the bottom arm is going towards minus infinity as your x is going towards minus infinity. In between, there can be peaks and valleys. By the way, these points are called stationary points in mathematics. Now what are stationary points? Need not worry about it right now, but these are the points where the slope of the tangents at these points become parallel to the x axis. Okay. Why does it call stationary point? We'll discuss all those things later on as of now. That is not our relevant topic because if I start talking about everything in the same chapter, this chapter will become humongous. Okay. Anyways, so please do not have this misconception that it is always having an inflection point like that, which I showed in x cube. It will have some peaks and valleys in between, but I'm not concerned about this part. I'm not concerned about this part. Okay. Many people in the last class, I talked about this in one of the sessions and they said, sir, how do you know at which point a valley comes and at which point a peak comes? But that calculus is required. As of now, don't worry about it. What is important is the domain and range for this. Okay. So look at this graph and comment on the domain and the range of it. Anybody, I would request you to write it on your chat box. What do you think is the domain and the range for this graph? What's the domain and the range? Very good, Achintya. Very good. How about others? Hi, Arya. How are you doing? Very good, Vihan. Now are you able to hear me? Is everybody able to hear me? Hey, thank you. I'm good. Thank you for asking me. I'm good, dear. Okay. So as you can all see, this graph has an extension from minus infinity to plus infinity along the x-axis and along the y-axis also it goes from minus infinity to infinity. So you can say that the domain of this function is all real numbers, all real numbers and range of this function is also all real numbers. Okay. You can either write an R or you can write minus infinity to infinity. Both are same things. Is it fine? Now, many people ask me, this is the story. When your A is positive, that means your leading coefficient is positive. How would this graph behave? Had the leading coefficient been negative? Let's talk about that as well. But before I move on, is there any question with respect to this? Okay. So many people will ask, what about the remaining part of the graph? See, I'll not draw the remaining part of the graph. As I told you, it could have various scenarios. It could be like this. No, it could be like this. It could be like this. Okay. So there can be so many cases happening in between, which is basically dependent upon your BC and to a certain extent on B as well. Okay. Right now, my main purpose is not to go into those details. I'll go into them when the right time comes. Okay. All right. So let's talk about what happens when your A is negative. Okay, let's talk about that case. So when your A is negative, that means the leading coefficient is negative. Please understand this. When your X becomes very, very large, your f of X will become negatively very, very large. Are you getting my point? Now, why is that so very simple? If this is negative quantity, A is negative quantity and if you increase your X, very large, very, very large. Because of this negative quantity, things will become very, very negatively large for f of X. Others BCD, they will not be able to save it. Okay. BCD, they have no, you can say dominance over this guy A. A is the boss. What A will say, they will have to follow. Okay. That is why it is called leading coefficient. So if A is negative and X is very, very large, f of X will become negatively very, very large. So it will become negative infinity. And Ulta will happen when your X is tending to minus infinity. So if X is tending to minus infinity, f of X will tend to plus infinity. Because of this, the graph of such kind of functions will look like this. Okay. So see, when X is very, very large, this graph will have an end arm down. Right. Signifying that when you are taking a very large value of X, Y will become negatively very, very large. And it will have one hand up. Okay. Something like this. You can see me on this, on the screen, right? Something like this. Okay. Other walls like this. This one is like this. Like this. Okay. So it will be dancing like this. So if your X is negatively very, very large, Y will become positively very, very large. Is it clear? Okay. I will show you some instances. Don't worry on the GeoGebra as well. Meanwhile, any question anybody has any questions? It's mirrored over Y axis. Not exactly Jatin. I should not say that mirror or Y axis. X is mirroring happens for some other reasons when your graph basically is, you can say unchanged when you remove your, when you change your X with a minus X, that type mirroring happens. In this situation, mirroring is you cannot see the word mirroring. That would not be an appropriate word over it. Okay. Anyways, just an example for this, I would like to show you the graph of negative X cube. Okay. Negative X cube graph basically looks like this. You have already seen this in our bridge course days. So it is like this. Yeah. So as your X is very, very large, it goes towards minus infinity. And if your X is negatively very, very large, it goes towards plus infinity. Okay. I will show this, these graphs in general as well. Any other question anybody has? Could you repeat which part if you want me to repeat? Regarding the orientation of the graph. Yes. So what I was saying, Arya, that if you have your leading coefficient as negative, if you had a leading coefficient as negative, if X is very, very large, the graph f of X value or the Y value will become negatively very, very large and vice versa. So because of that, the orientation of the graph will be this. In between some kitschery will happen, which I am not, I will not comment on this because it depends upon DC and D. So in between some, you know, activity might happen, which depends upon DC and D also, which I'm not commenting right now. Okay. So let us look into one of such graphs on GOG graph. Now same graph, same graph, I'm going to make a minor change. Instead of an X over here, I'll put a minus X. See the, see the nature of the graph, right? The moment I press and enter, see what will happen. Do you see this? Let me drag it a little bit up. Okay. Do you see this graph? So if you see this graph as X becomes very, very large, this guy goes to minus infinity. So when X is very, very large, it goes to minus infinity. And when X is very, very negatively large, this goes to plus infinity. Okay. So opposite trend has been shown by it. In between some peaks and valleys are there. But as I told you, I am not going to talk about it. What I'm going to talk about is what is the domain and the range, because that is our subject matter of study in this chapter. So what do you think is the domain range for this graph? Tell me, tell me fast, fast. Domain range, domain range. Bega, bega. Right, Achindya. Absolutely. Very good. Awesome. So in this case also, you would realize that the domain and the range will be all real numbers. No difference. No difference from the case where your leading position was positive. Okay. So domain all real numbers, arrange all real numbers. Right. So what is the conclusion over here? The conclusion over here is that any cubic polynomial, if a question comes in your school exam, to find the domain and the range blindly, close your eyes, not even have to look. Right domain is all real numbers, arrange as all real numbers. Right. Of course, write it against the proper question. Okay. So if the question center says, find the domain, find the domain and range for something like minus seven X cube, plus six X square, plus two X, minus 100. Okay. Now, don't worry, cubic polynomial, right, domain is all real numbers. Okay. So don't have to even waste a single moment. Domain, domain will be all real numbers. Range, range, all real numbers. Over. Finish. This is your answer. Okay. Don't have to worry whether, whether leading coefficient is positive, negative, whether there are other terms or not. Domain range will always be minus infinity to infinity, respectively for both of them. Is it fine? Any question anybody has? Okay. Later on, we will find something more interesting. Okay. I'll add something more interesting to it. Now, can I go on to the next polynomial? Any questions here anybody has? No doubt. Great. So now let's talk about the bi-quadratic polynomial. I'll be slightly faster here because we have already done this. It's all about recalling bi-quadratic polynomial. Degree four polynomials are called bi-quadratic polynomials. So a typical bi-quadratic polynomial will have something like this, AX to the power four, BX cubed, CX square, plus DX, plus E. Okay. Of course, when in the bridge course I was discussing about bi-quadratic polynomial graph, I never made a mention of such a heavy one. Basically, I dealt with a very pure form of a bi-quadratic just having X four term, but it is quite possible that a bi-quadratic polynomial can have other terms like X cubed, X square, X and constant in them. Okay. Now, again, the leading coefficient plays a very, very important role. What is the role it plays? If your leading coefficient is positive, then the graph will show the following characteristic. What will it show? If X is very, very large, as you tell me now, f of X will be just write INF or minus INF, infinity or minus infinity. What do you think? What do you think? Very good, Jatin. Very good, Achyentia. Awesome, guys, you are superb. Okay. It will go to plus infinity, very good. And as X goes to minus infinity, Y will go to, Y will go to, Y will go to, Y will go to, Y will go to minus infinity. Very good. So, this tells you that one second, one second, one second, one second. It will still go to infinity. See, you have raised this to a positive, I mean, even power. Right? So, if you put minus infinity, even power will convert it to a positive quantity, right? So, everybody got carried away there, right? So, what does it tell you that? It tells you that irrespective of whether you are going towards plus infinity or minus infinity, the graph will always go up. Up means plus infinity, right? So, the end arms of the graph, my dear, would always be facing upwards as if like pointing towards six. The way the cricket umpires say six. So, six, okay. So, in between something will happen. Again, I'm telling you, I'm not going to comment about what will happen in between. In between, some peaks and values will happen. But maximum, you will have three sets of peaks and values, okay. In between and also. So, in between something will happen. But what I'm more interested in is that its end will be towards if it goes to a very large value of X, Y value will also go very large. And if it goes to a negatively very large value of X, Y value will still go very large, okay. In between something will happen. In between something will happen. I mean, I'm not in a state to comment about it till I know the value of B, C, D, E, etc. Okay. So, this is very interesting observation. So, this goes to plus infinity. This will also go to plus infinity. And if this goes to minus infinity, this will also go to plus infinity anyhow, okay. Now, I will show you some instances of the graph. Let's look into some graph from GeoGebra. So, let me show you a graph of GeoGebra. Now, same graph. Let me just make a minor modification. Let me remove this and let me put a power of four over it. Ayo, power of four. Yeah. There you see. So, let me drag it down. So, what you see is it has got a unique kind of a valley and a peak over it. And you can see here that the graph is always going towards positive infinity. Irrespective of whether you are going towards infinity or minus infinity on the x-axis. Okay. So, the two arms are always up. Are you getting my point? Now, when you're given such a graph and somebody says, hey, what do you think is the domain of this function? What will you say? Domain means what is this fan? Achinde, you're partly correct, but not completely correct. Exactly enough. Exactly, Arya. So, domain for such kind of function. So, let me go back to my black screen. Domain for this function is going to be all real numbers. Okay. But what about range? What about range? Now, for range, my idea is, let's go back to our GeoGebra graph. Range for this graph will be from this point onwards, this point onwards, whatever is this value? Okay. All the way till plus infinity, correct. And this value can only be found out. This is the minimum point of the graph. This value can only be point found out by calculus. So, calculus is required for it. No, no, no, no. In the present example, which I have taken, the range is probably from minus five to infinity. Okay. But minus five because you're seeing the graph, right? If you have not been, I mean, if you have not used any kind of a mathematics on that and nobody has told you any graph of it, range will be difficult thing to find out in such cases. Please remember. Please remember. No, no, no, it is not all positive real numbers. Again, let me tell you here, this particular example itself violates that it is from negative five to infinity in this case, but it depends on case to case. So, what is going to happen in case of a bi-cordetic polynomial? Listen to this very clearly. A bi-cordetic polynomial will have a minima somewhere. It will have a minimum point somewhere like the one which we saw in the GeoGebra graph. Okay. And for this, we require calculus. Okay, calculus is required for this. You cannot find it out without the use of calculus. You have to find dy by dx. You have to put the slope to zero. We have to solve that equation and get our position of the minima and the value of the minima. So, for range, you need calculus. So, it will be from some minimum value to plus infinity, but for this minimum value, we need calculus. Okay. So, in your school exams and your symptom exams, we will not be asking you to find the range of a bi-cordetic polynomial other than the few exceptional cases. For example, x to the power 4, you can answer. For x to the power 4, the minimum point is zero, maximum point is infinity. So, it goes from zero all the way to infinity. But in general, we will not give you a complicated or we will not give you a normal bi-cordetic polynomial to find the range because without calculus, you cannot find it out. Okay. Yes. If you see the graph, if you see the graph of the bi-cordetic polynomial, I'll just show you back again. There are two values here. Right? So, as you can see, there are two values over here. One is here, another is here. And there is one, you can say, peak over here, this point. Okay. So, how do you find the position of the lowest value without calculus? You cannot do that. Okay. Both are minimum points only. Both are called local minimum points only. But without calculus, we will not be able to find it out. So, to answer your question, sir, if there are two values or dips, both have different minimum, but yes. So, we have to see which is the more minimum of the two, which is the lowest of the two. And then we have to decide the range of the graph. So, a good news is that we will not be asking you, in fact, in your school also, they will not be asking you for the range of this. Okay. Because you require calculus and calculus has not been taught to you. But don't worry, as when the calculus will be taught to you, you will not be spared. Questions coming, questions based on finding range on bi-cordetic polynomials, then we'll start coming for your exams and tests. Okay. So, this is the story when your leading coefficient is positive. Let's talk about when your leading coefficient is negative. The analysis is more or less the same. In fact, you will be also able to do this analysis on your own. So, when your leading coefficient is negative, when your x becomes very, very large, remember f of x will become negatively very, very large. And same will be holding true even if your x is negatively very, very large. So, whether your x goes to infinity or minus infinity, because the leading coefficient is negative, a x to the power four will always be a negatively large quantity. This basically indicates that the way the graph will be oriented will be something like this. The end arms will be facing downwards like this. In between some peaks and valleys will happen. So, in between some peaks and valleys will happen, which basically I will not be in a position to comment till I know the exact function. Okay. So, what I want to say here is that when x becomes very, very large, y will become negatively very, very large. And x becomes negatively very, very large, y will still become negatively very, very large. Okay. Let me show you one of the cases on one of the cases here. The same graph, now I will take my x leading coefficient to be a negative number. Maybe let's say negative two. Okay, there you go. See the way this graph has behaved. By the way, I should not put such a large number because it is killing away the valleys. Maybe I'll just put a negative sign. Let's see what happens. Okay, it's just taken a turn. Okay, I'll just give another version of it. I mean, the agenda is fine. The agenda is fulfilled that both the arms are going towards minus infinity, but I will try to put a better picture over here. Let's say I'll put a plus sign. Okay. Yeah. See this. So what is this graph trying to show you? This graph is trying to show you that as x becomes positively large, this function goes down all the way till negative infinity. Okay. And when this value of x becomes negatively very, very large, still it goes to negative infinity. Okay. In between there are some peaks and valleys. Now, what do you look at the graph and tell me what is the domain of the graph? You will say the domain of the graph is no different. It is still minus infinity to infinity. But what is the range of the graph? You say for range, I need to know this value. Right. So this is the max point of the graph. Min is anyways negative infinity. Right. So in such case, to find out the max, you need calculus. Calculus is again required. Without calculus, you will not be able to move even a single inch in this particular concept. Okay. So coming back to this, please note that the domain in this case, the domain in this case. What happened? Yeah. Domain in this case, just a second, guys, I think I need to change my digital pen. It is giving me a lot of issues. So I'll just get a different one. So domain here will be again, all real numbers. Oh, what is this? Yeah, all real numbers, or you can say minus infinity to infinity and range here would be from a max position or you can say minus infinity to a max position. And this max position, please don't start writing max in your paper. Right. Because this is something which will be coming out from calculus. So you require calculus for it. No, no, see, don't worry. Your teachers will not ask you for a range question based on this. Maybe they will ask only the domain question. Okay. So even I'm not, I know, talking about what will be the max position over here. Okay. So one of the peaks will be the maximum position depending upon which is more. Okay. So in this case, you would require, you will realize that the conclusion is the summary of this is for any, let me conclude it for any by quadratic polynomial for any by quadratic polynomial. The domain will be always domain will always be minus infinity to infinity. And range would be, please note this down, if is positive, it will be from a minimum position to infinity. If a is negative, it will be from negative infinity to a maximum position. Okay. Where minimum and maximum values can be figured out from calculus where min and max are obtained from the use of calculus. Okay. We cannot find it without, you know, we cannot see sketching is anyways a difficult task. Okay. Even if you want to apply mathematics to get it, we cannot find it without knowing calculus. Is this fine. Now I can go on and on I can go to a pen take I can go to a degree six degree seven degree eight degree nine but let me conclude here about polynomial. So what we have learned about polynomial is the following things. First of all, have you all copied this down. Any question that that it would like me to answer here. All fine. Yes, please copy it down. I'm giving you time. Just let me know done once you're done. Okay. All right. So let's do a quick recap. We started our discussion with in the polynomials from a constant polynomial. So a constant polynomial was basically a horizontal line extending all the way from minus infinity to infinity. The domain was all real numbers. Correct. Keep an eye on the domain and the range range part. Okay. And range was a singleton set, which was equal to the value of the constant. Right. So f of x equal to three range is a singleton set three. Then we extended our learning to linear polynomial linear polynomials are basically like lines, which extend all the way from minus infinity to infinity, both along the x axis and along the y axis. So the domain was all real numbers range was also real numbers when it came to quadratic domain was all real numbers, but range was from a minimum to a maximum position of infinity. That means minimum vertex position to infinity if your a was positive and maximum position to minus infinity if a was negative, and that maximum minimum positions where nothing but the y coordinate of the vertices, easily found out by use of the formula minus D by 4A. Right. Remember the last class. Today I started talking about cubic polynomials here also we realize that domain and range both will be from minus infinity to infinity. But when it came to by quadratic again we basically got domain as minus infinity to infinity but for the range we got the minimum maximum problems. Okay. And these minimum maximum things that I can see on your screen, they're obtained by the use of calculus. Right. So the trend here or the pattern that you would notice here is that. So let's do a conclusion with respect to polynomials for any polynomial. Let me write it down for any polynomial or any polynomial. The domain is the domain is all real numbers. The domain is always all real numbers. So you can try you can try any polynomial existing in the world domain will always be all real numbers that means polynomial are such machines, which you can feed anything. Okay, they will eat anything they want. Okay, so whatever number you want to feed into a polynomial they will be more than happy to accept it. Right. But when it comes to range, but when it comes to range. This is a second guy that thing. Can you all see me visible to you all. Okay, because now I have to. There's just a small power to just give me a second. Yeah. Yeah. But when it comes to range. When it comes to range. Please note that for odd degree polynomial for odd degree polynomial range is given as all real numbers again. But for even degree polynomials, but for even degree polynomials. Your range is given from the max to the minimum position. If you're leading coefficient is positive. So if you're leading coefficient is positive. Then range is going to be range is going to be from a minimum to a infinity. And if you're leading coefficient is negative, your range is going to be from, sorry, from minus infinity to a max position. The only exception to this, even that is not an exception, the only even degree polynomial where you can find this min and max easily is your quadratic polynomial. Right. So let me write down here. For quadratic polynomial. Min and max are easy to find out. Right, because that doesn't require you actually to go to calculus. Okay, so for quadratic polynomial. Your min will be and max will be minus D by 48. So for this, you do not need the use of calculus. You will use calculus. You will need calculus when you have a higher degree polynomial like degree for degree six degree eight for quadratic you for quadratic you don't need a calculus. So this page beautifully summarizes domain and range concept for any polynomial function existing in this word. Let's say number one, it says that for any polynomial domain is always all real numbers blindly you can say without any second thought. But when it comes to range, it actually depends whether you are dealing with odd degree polynomial or even degree polynomial. So for odd degree polynomial range is also all real numbers. When it comes to even degree polynomial, you need to basically see what is the leading coefficient is it positive then it will be some for minimum value to some infinity. If it is leading coefficient is negative it will be going from minus infinity to some maximum value where this minimum and maximum value are found from calculus. So use of calculus is required. The only gentle case, the only sober case I would say the only simple case I would say in such cases your quadratic polynomial where you do not need any calculus min and max can easily be obtained by this formula minus D by 48 D is the equivalent everybody knows about it is this second in your mind because this is going to be for there with you for the next two years. If you have any doubt any question, please get it at this now itself. Okay. Understood. Now, let me ask somebody to summarize whatever I said. Any volunteers by the way, let me not take the name any volunteer who would like to conclude this polynomial domain range concept. Anybody Harsita would you like to conclude this. Would you like to conclude this to unmute yourself. Yes. Yes, please go ahead. Yes, so for any polynomial, the domain is the set of all real numbers. Very good. And for range, if it's an odd degree polynomial, then the range is a set of all real numbers. Very good. And then even degree polynomial if the leading coefficient is positive, then the range is from a minima to positive infinity. How will you find this minima? Using calculus which we'll deal later on but for a quadratic you know how to find it. Yes. Very good. Go ahead. And if the leading coefficient is less than zero, that is negative, then the range will be from minus infinity to a maximum. Beautiful. Thank you so much. Thank you so much. Okay. Is it clear now everybody? Fine. So with this now we will be moving on to rational functions. So in our list of functions, if you recall, in the last class I had given you that we will be discussing few types of functions in our discussion. Under algebraic function, we have successfully finished off polynomials. Now we are going to talk about domain range. And here and there we'll sketch graphs also. But graphing is not very important as of now as domain and ranges. We are going to talk about rational functions. Rational functions. What are rational functions? Does anybody remember? What is a rational function? So when do you call a function as a rational function? Any idea? Anybody? That is an example of a rational function. But see if I say, what is a human? What is a human being? A kill. I don't have a human being, but there can be many more human beings. So how do you define a rational function? Rational function is nothing but a function which is expressed as a polynomial divided by another polynomial. So here P and Q both are polynomials only. Okay. Let me give you a few examples. What happened to my polynomials? Yeah. So some examples which I can sign for this. Easy ones. One by X is what REI is saying. So as you can see, one is a polynomial, degree zero polynomial. X is a polynomial, degree one polynomial. So this could be an example. You could have X minus two by X plus four. This could be an example. You could have something like X plus one by X square minus three X plus six. I mean, I'm just taking some polynomials here. So these are all examples of polynomial function. Okay. So now we are going to start talking about the domain range. And in some cases, we'll be also looking into the graph, but graph is not very important for us. Okay. Let us start with, let us start with. Polynom rational functions, which are made up of, which are made up of. So let me take a few cases for you. The first case I'll be talking about such polynomial, such rational functions, which are made up of a constant. Let me write something like constant divided by a linear polynomial. Okay. And to a certain extent, we will also see their graphs as well. So let's say I take this example, maybe two divided by three X minus five. Okay. Now, if you remember your transformation of graphs concept, which we learned in the bridge course. Let us try to make the graph of this particular rational function. And when we have made the graph, we will also look into the domain and the range of the graph. Okay. And then we'll try to come to some particular theory related to it. So for this particular function, if you see, let me initiate with my basic graph of f of X is equal to one by X. Okay. Now, all of you please pay attention one by X graph. You had already done in your bridge course. It's going to be a rectangular hyperbola like this. Okay. So these are the two arms of the same graph. These are the two arms of the same graph. Okay. Now, all of you please note that at the back of the mind, I have this particular thing running on which I have not explicitly written over here. So if I write this function like this, everybody's fine with that. You can simplify this, you'll get the same answer back. Okay. So at the back of the mind, I need to reach this destination. So for reaching this destination, I will do some set of transformations. Okay. Right. We'll come to that. Okay. Now, everybody please pay attention. So this is the graph of one by X. Now, if I ask you how would the graph of two by three divided by X will look like or two by three X will look like. See, the multiplication of two by three just tries to scale the graph by a factor of two by three position wise, it is not going to get disturbed. It is not going to move right or left or up or down. Okay. So I'll just show you how how two by three will affect the graph on GeoG graph so that you get a good idea about it. So I'm not going to mute this graph and I'm going to make Y is equal to one by X. Okay. This graph, everybody can see correct. There you go. Yeah. Now just watch carefully. I'm not going to multiply. I'm not going to multiply two by three to this. So two divided by three X. There you go. Okay. Not much of a change. No shifting here and there. It's just going to scale up on the same place. Okay. At the very same place, it went to scale down in fact, scaled down by a factor of two third. Okay. If I go to this graph, multiplying with two by three doesn't have a substantial, you know, change in the shape of the graph. So maybe it is slightly more push. Okay. It will push go five more. Okay. Not much of a change. Okay. Now everybody please pay attention. Now the third step I'm going to replace my X with X minus five by three. Now for the people who were not in the bridge course with us, I hope when you're looking at it, you may be finding things slightly unknown to you. Okay. In that case, please ask relevant questions. Okay. So now who will tell me, I will tell me when you change X with X minus something and that something is positive. What happens to the graph? But I tell me moves to the left. Are you saying moves to the left? X is positive and negative. X is positive moves to the left. Just give me a second. I think my volume of my speaker is, yes. Does it move to the left or does it move to the right? Right. It moves to the right. Yes. By five by three. Exactly. It moves by five by three. Very good. So what will happen? This arm, this arm will move five by three units like this. That means it will become in this shape. Okay. Please note that this means your vertical asymptote will now become X equal to five by three. Right. Now looking at this graph, you tell me what should be the domain and what should be the range of this particular function. You say sir domain. I can see the graph extending all the way from minus infinity to plus infinity. The only one value it doesn't take, which is five by three. Right. Somebody was saying, sir, it cannot take five by three. I think Satyajit absolutely correct. So this guy can take all real numbers except five by three. Are you getting my point at five by three, this function will be undefined. It doesn't know what to do because your denominator will become zero. As you can see this guy will become a zero. Are you getting my point? Now what about the range? If you see the range here will be all real numbers except zero. As you can see your Y value is not becoming zero. It is stopping below zero and starting from slightly above it. See all the way from minus infinity is coming, coming, coming, coming, stopping just before zero. Before zero means it is not touching the X axis. And then again, it's starting slightly above X axis and going all the way to plus infinity. Yes or no. Okay. So this is the domain and the range of the graph. Now many people ask me this question, sir, does it mean I have to draw a graph every time to find domain and range? No. I will tell you now a way by which you can solve this type of question without sketching a single line. That means domain and range can be found out without having you to sketch the graph. Okay. I'm sure most of you would have done this in school as well. So we will take that approach in the next two minutes. But before that, any question anybody has, please do ask because I'm going to go to the next screen. Any questions? And then we'll take a few questions based on that. Don't worry. Clear. Any questions, any concerns? No doubt. Now, Arya, try speaking something. I have increased the volume of my system. Hi, sir. Hi. Okay. Now we will move on to a theory based on this. Okay. So how to find domain and range for a function of this nature, a constant divided by a linear polynomial. And later on, I will complicate it. As you know, my nature is to complicate things slowly. Okay. I will never do it initially. I'll start with very basics and then I will complicate it. Okay. Okay. All of you, please pay attention. When you have, let's say the same question. When you have this function, f of x or y equal to, I mean, it doesn't matter whether you say f of x or y, the function was two divided by, I think, 5x, sorry, 3x minus 5. If I'm not mistaken, 3x minus 5. Okay. Now, please note in such cases, our domain is domain is all real numbers except those values of x, which will make the denominator go zero. Okay. In this case, your denominator is 3x minus 5. Okay. So in general, if I have to write it, in general, if I have to write it, so if you have, let's say a constant divided by q of x, so your domain is going to be, your domain is going to be all real numbers except, except minus sign shows except. Okay. Except those values of x, which will make the denominator polynomial become zero. So whatever values are making your denominator polynomial zero, remove them, remove them from all real numbers. So whatever will be obtained, that will be a domain. Got the point. So this is without using graph. Again, straight away go to the domain and you can answer this question. So here, if I try to solve this question, then basically I will be removing from all real numbers that value of x, which will make 3x minus 5 becomes zero. So which value makes 3x minus 5 becomes zero. You'll say, sir, 5 by 3. So remove 5 by 3. Then this is my answer for the domain. Clear any question, any concerns, do let me know. No doubt. Okay. Now what about range? Guys, few things before I would, before I start finding about range. Number one, finding range is slightly more complicated than finding domain. That thing you have to, you will learn in, you know, after solving a substantial amount of question. Number two, without domain, there is no existence of range. The second point is very important. I've seen people directly jumping for finding the range. Even if the question says, find range, no domain, you have to still find the domain. Get this clear. Even if the question says, find range only, you still have to find the domain. There is no existence of range without domain. If you don't know your input, how would you decide your output? Range is what? Range is output. Domain is input. If you don't know the input, how would you know the output? Are you getting my point? This is a very, very strong statement which I'm making here because the last 12, 13 years of my teaching experience, I have seen people misusing this rule left, right and center. If you don't know the domain, you go straight to the range. No. First domain. Then when you know your domain, then accordingly, you will act to find out the range. I don't know how it has been taught to you in the school, but I personally am a very strong supporter of this concept that without the domain, there is no existence of range. Domain decides what is the range. For example, I'll give you a very simple instance. You may find it very trivial. Let's say I give you this question. There is a function which is defined like this. Okay. Very simple question. Tell me the range of this function. So this function works from this set to all real numbers. Tell me the range of this function. Can you write it down on your chat box? Tell me the range of this function. I know I'm asking you a very simple question, but still I have a feeling there is a, you can say, hunch that somebody is going to give me a wrong answer for this. Right, right, right, right. Nobody is writing. Everybody is quite excellent. As expected, I mean, as expected, you have got it right. But trust me, many people will get it wrong also. I mean, God forbid it should happen. Everybody should give me the right answer. You know, 99% of the Janta initially will say, sir, it's a polynomial of degree one. Range is all real numbers minus infinity to infinity. No, watch out the domain. This is the domain, my dear. How can you expect an infinity to come out? How can you expect a hundred to come out when your inputs are only limited to two, three, four? Are you getting my point? So if you have a blind eye towards the domain, you will not be able to find the range correctly. Let me tell you. So in this case, as rightly said by Jatin, range will be the values of f of x only for these three values. So when you put a two, this will give you a three. When you put a three, you will get a five. When you put a four, you will get a seven. That's it. This is your range. Over. Story is over. Okay. This is your answer. Don't start writing all the numbers just because there was a theory given to you that for linear polynomial or a odd degree polynomial, your range will be all numbers from minus infinity to infinity. That is fine. That is under the assumption that your domain is minus infinity to infinity. But if I cut short my domain like this, range will also get cut short. Other of machine code, two pillow, you have three Dal or four Dal only three. These, these three things can be put into the machine. Machine will accordingly give you the answer. No. How will you machine give you five lakh and one code and all. Right. That's that's the, you know, so many people have a blind eye towards this fact. So please be careful. Anyway, coming to a range part. So range has no existence without domain. So how do you find range in this case? No, for range, what I do is whatever is the function given to me, I will rewrite it once again. I will make X the subject of the formula. Why I make X the subject of the formula? Because X is your input and I know everything about the input. Okay. So what I'm going to do, I'm going to write X in terms of Y or I'm going to make X the subject of the formula. How to do that? Very simple. Just multiply this to the left. Okay. Take your, take your terms like this on the other side. Okay. I hope I have not. Yeah. Yeah. So your X will become this. Okay. Your X will become this. Right. Any doubt? Till this step. No doubt. This should, this is just a simple, you know, equating it to Y and getting X in terms of Y. There should not be any doubt with respect to this. Any doubt, any questions, please do let me know. No doubt. Any question anybody has. Now, what is the restriction on the domain? The restriction on the domain is your X should be all real numbers except, except five by three. So can I say, since this is the restriction on X, the same restriction should be on this guy as well, because this is a representation of X only. So this guy should also be all real numbers except five by three. Yes or no. Yes or no. Since both are, since both are representation for X only, whether you write X or whether you write two plus five Y by three, if your X belongs to this interval, so should two plus five Y by three Y also belong to the same interval. Okay. Now, this guy will anyways not take five by three. Why? Let's check. Let's check. So in the side, I will make a small calculation. I will try to put this as, I'm sorry. I'll try to put this as five by three. See what will happen. Okay. When I do that, I realize that it will become two plus five Y is equal to five Y, which means two is equal to zero, which is not possible. So this guy anyways will not become five by three. That is for sure. Okay. You may try to test it out. You may try to put it as five by three. You will get a shock of your life. You will see that Oh, two is becoming equal to zero, which is not possible. But all I want to check is that this guy must be a real number now. So if you want this to be real number, the only restriction that you need to honor over there is that this denominator should not become zero. Exactly. So if you want this to be a real number, it means your three Y should not be zero or your Y should not be zero. That's it. Yes. Any value you put it is fine with it. It is happily take any value of Y put five, put hundred, put negative two lakhs. It is happy to take those values. But what it is scared of is don't put a zero in Y because the denominator will vanish because of that. The denominator will become zero because of that. So hence, we say since Y cannot be zero and why is what? Why is the output? Why is the output right? Output is a range. So your range cannot be zero. So what we say is that range can be any real number except zero. So here you see I have found out the answer for domain and range without making any graph. Is this fine? Is this clear? Any questions here? Anybody? Any questions? Okay. So should we now scale it up? Let us scale this concept to talk about those rational functions which have a linear by linear. Yes, yes, yes. I'll give more examples already. You want of the same time? Let's do of the same time. We'll take one more of the same time. No problem. We'll take one more. So let's do a small question. Find the domain and a range of this function. Let me put four upon three minus two X. Everybody do this question and give me a response on the chat box. Excellent. Domain is very easy to find out. Yeah. Correct Vishal. Okay. Very good. Very good. Very good. Very good. So can we have given both the answers? Others also please do this guys and girls. Let me tell you this concept is very important for your school as well. Right. Everybody wants to do well in school exams at least. Right. UT semester exam. These type of questions definitely will come. Definitely very good. Okay, Justin. Should we discuss it now? it now anybody wants more time domain not exactly area you made a small mistake see for domain as I already told you let's discuss it everybody some of you are not very confident so I'll just do it once again domain is all real numbers except those values which will make the denominator become zero okay so if you solve this part if you solve this part you'll end up getting 2x is equal to 3 so x is equal to 2 by 3 so it means your answer will be all real numbers except sorry I made a small 3 by 2 sorry 3 by 2 so this will be all real numbers except 3 by 2 okay not zero got it okay so this will become the domain of your given function this will become the domain of your given function no worries with domain everybody got it right domain is the easiest part okay domain is the easiest part now coming to range now that I have known what is my domain so I know what all I can input in my function let's try to figure out what is range range means what is the value that y can take right so if I equate this to y so what is the what is the extent I can choose values for y that is an output so here again I will make x the subject of the formula so basically this is a simple operation so I can say 2xy is equal to 3y minus 4 so x is equal to 3y minus 4 by 2y now since your x is all real numbers except 3 by 2 it implies it implies 3y minus 4 by 2y should also be all real numbers except 3 by 2 correct yes or no but 3 by 2 anyways it will not take so there's no I mean this is not a relevant thing for it because see it cannot take 3 by 2 try to try to equate it to 3 by 2 I'll just do it over it if I equate it to 3 by 2 you would realize you'll end up getting seeing something very weird which means minus 4 is equal to 0 not possible okay so anyways this guy will not be 3 by 2 so only thing I need to take care is that it should be real if you want this to be real this guy should not be 0 that means y should not be 0 that's it so this means the range of this function is all real numbers except 0 same answer as what we got for the previous one same answer as what we got for the previous one is this clear and everybody those who made a mistake is it now fine with you okay now let me take let me go to a slightly more complicated case so I will our concept will get more refined as we take more complicated cases is this fine everybody any questions so domain is everything except 3 by 2 range is everything except 0 so if you see the graph I will draw it over here for you let me move to these guys so I think my function was y is equal to what was it 4 by 3 minus 2x 4 by 3 minus 2x 3 minus 2x there you go so you can see that x equal to 3 by 2 will become its vertical asymptote there you go that means this function will never become a 3 by 2 the moment it tries to take 3 by 2 it will become undefined okay and range if you see it's coming all the way from minus infinity to little below 0 and then from little above 0 to infinity right just the for the domain just the denominator should not be 0 we will we will reach that conclusion in some time as I take more and more situations okay so now I'm going to take up a case I'm going to take up a case where you have a functional equation sorry you have a rational function I was teaching functional equation to your seniors and I started using function equation so now you have a rational function where you have a linear polynomial by another linear polynomial by another linear polynomial okay so basically it's a case where your t and q both are linear polynomials only let's take an example let's say something like this x minus 1 by x plus 3 okay so you can see here in this case both your numerator polynomial and denominator polynomial are linear functions or linear polynomials okay now how do we find domain in such case now for the domain the concept is no different so let me call them by the name of p of x and q of x and both of them are linear only so both of them are linear polynomials only so in such cases also the concept of domain is no different so your domain in this case will be again all real numbers except those values of x which will make the denominator go zero simple as that so in general in general if I have to write it okay domain is going to be all real numbers except those values which will make the polynomial below that is a denominator polynomial become zero so nothing different from what we did in the previous case nothing different same thing same thing here also okay because see the both the guys are polynomial right they don't have any problem taking any input we have just now seen polynomials are functions which you can feed anything you want right they are not perturbed by any input you give give it to them but they're only as a team they're only perturbed when you give something which makes the denominator become zero because if that happens the whole thing becomes undefined so if the base becomes zero okay so it'll become undefined expression so they're fine give anything to them okay I'm happy with anything but don't give me anything which will make the denominator go zero because then this expression will fall down is this fine okay so in this case when you solve this question your answer will be the answer to this question will be all real numbers except except negative three okay this is going to be the answer for the domain now range range also the idea is more or less the same okay nothing will be different much so for range same thing we will first try to write x in terms of y so what I'm going to do I'm going to write x in terms of y same approach so multiply it okay so x y plus 3 y x minus 1 let's make extra subject of the formula and if I'm not mistaken x will be 1 minus y is equal to 3 y plus 1 so x will be yeah x will be 3 y plus 1 upon 1 minus 1 okay now since x belongs to all real numbers except negative 3 it means same will be true for this guy also okay this guy also will be all real numbers except negative 3 but negative 3 anyways won't take why let's see negative 3 anyway this guy won't take because if you equate this to negative 3 see what will happen you'll get the shock it becomes negative 3 plus 3 y so that means you're trying to say 1 is equal to minus 3 which is not possible so this guy anyways won't take negative 3 right because that is not mathematically possible but what it should take is a real number other than minus 3 so if you want this guy to be real number if you want this guy to be real number what is one condition that you need to fulfill think as if you're finding the domain of this function itself so what is the domain of this guy itself you say why should not be 1 exactly I will say sir it is fine with anything you feed to it but it will not not be fine the moment your y is such that your denominator becomes a 0 that means y cannot be a 1 right and indirectly you have basically written your range so your range will be all real numbers except one is this fine so this is your range this is your domain any questions here anybody any questions okay would you like to see the graph also would you like to see the graph let's see the graph could you scroll a bit to the other side other side means to the left of the screen this way other way other way we are at the left left most part only I think this contains the most amount of information done okay we'll see the graph also for this let's see the graph yeah can somebody dictate the function to me what was it anybody can x minus 1 by x plus 3 x minus 1 by x plus thank you there you go okay now here if you see this function this function will take any x other than minus 3 so y sorry x equal to minus 3 will be like a vertical as you go see it got it and this guy will take any value other than 1 so y equal to 1 output can never be achieved as you can see y equal to 1 is a horizontal asymptote okay is it fine jatin has a question in f of x can we say that the value for the numerator is 0 is the range the value for which the denominator is numerator is 0 is the range no not necessarily no sir that was the typo sir actually what I was saying was like it's not included in that value like if you take the example x minus 1 by x plus 3 now x minus 1 is there effects in the numerator is 1 then it becomes 0 right okay maybe in this example it is trying to give you a feel like that but uh so one shouldn't be included in the range that's what i was saying when the denominator is 0 won't be included in the domain because while we are making it in terms of y what we do is cross multiply right so the numerator of the f of x becomes the denominator when we take it in terms of x we'll see more examples maybe let's say in the next example we get some kind of hint like that we'll see okay we'll see that particular aspect of it but uh actually it is not that actually it's the limit of this which it cannot take you have you have learned limits right but you have not learned a case where the limit is when your x is becoming very very large when your x becomes very very large this answer is going to be a 1 that value it cannot take because x becoming very very large itself is not possible because you're trying to say x becoming infinity infinity is not achievable that value becomes that value is removed from the range other than the other right now you're seeing here minus 1 right if i make it 2 then range will not include 2 if i make a 2 over here if i make this guide 2 let me just write it i will remove it after this then your range will not include 2 let's take an example let's take the same example find the domain and range of this function as i told you i will put a 2 okay let's everything is same uh denominator was i believe x plus 3 only right okay let's try to solve this in fact i will also solve along with you so that we don't uh you know devote a lot of time for this question so for domain it is pretty simple domain you will say sir all real numbers accept those values of x which will make the denominator vanish right so only minus 3 makes it vanish so the outcome of it is x equal to minus 3 so from all real numbers remove minus 3 that will be your domain so this is your domain answer no doubt about this everybody is fine with this isn't it now let us find out the range for range processes same as that what we took in the previous example okay let's cross multiply okay and uh let's make x the subject of the formula okay so x becomes 3 y plus 1 upon 2 minus y clear now since your x is all real numbers other than minus 3 since this belongs to all real numbers other than minus 3 it implies even this guy should be all real numbers other than minus 3 correct now minus 3 anyways it won't take that we already know i don't want to waste time doing the same thing again and again but if you want it to be real at least you need to ensure that this guy is not zero that means y should not be two so what did i say ajatid so your range will your range will not include two into it it's not about the numerator point becoming zero right so uh this case is basically linked to your understanding of limits which uh still not uh uh you know to the requirement we will we will complete our limit chapter officially when we started in our regular program after that you will be able to make sense out of it is this fine okay now time to complicate it even further right time to complicate it even further and now what will happen if i put linear on top and quadratic in the denominator okay while domain is easy for that range will require another important prerequisite for it and what is that prerequisite i will talk about it before i give you a you know any question on this so now we're talking about domain and range for such or such rational functions where the denominator where the denominator is a quadratic okay and numerator could be linear or constant anything linear or constant okay but this has a prerequisite before i start talking about pre-requisite before i start talking about domain and range concept domain is easy domain i'm not you know this prerequisite is not for domain this prerequisite is for range okay so before we take a question on range there is a small prerequisite i should not say small this is actually a substantially important concept for us so that small prerequisite is you must know wavey curve sign scheme also called method of intro okay i'm sure your school teachers would have done this okay whether you were aware of it or not she definitely would have done it okay because without this you will have a problem in solving at least you know some questions based on domain and range with respect to this now what is this concept have you done this wavey curve sign scheme in school anybody nafflers would have done it what about raja ji naga students anybody remembers lately your teacher making a number line putting plus minus signs on it any kind of approach like this was done in school yes no maybe yeah yeah now you remember okay what she did was actually this guy wavey curve sign scheme so what is this sign scheme let us try to understand this okay now wavey curve sign scheme basically is the method this is the method which helps to solve a rational function inequalities okay so this is the function by which we can this is the method by which we can solve something like this i'll just draw and just write a generic expression x minus alpha 1 x minus alpha 2 da da da da till x minus alpha p and let's say they have some powers k 1 k 2 da da da k p divided by x minus beta 1 x minus beta 2 they may have some powers also i will let it down let's say l 1 l 2 till l q okay and it helps you to solve these inequalities like greater than equal to 0 or greater than 0 or less than equal to 0 or less than 0 okay any of these four type of inequalities involving a rational function to the left and 0 on the right can be solved by wavey curve sign scheme okay to explain this i will take an example it is best understood by an example so how does wavey curve sign scheme help us to solve such kind of examples such kind of questions okay so let's take an example let us say i want you to solve this inequality i hope i'm not taking very heavy figures such a high high figure you're taking let's say i make it to the power of 11 okay let's say i want to solve this inequality okay what is the meaning of solving an inequality what do you understand when somebody says solve this inequality it simply means for what values of x or for what interval of x will this particular inequality is met okay so for this i'm going to use wavey curve sign scheme also called as methods of interval listen to this very very carefully because because it is not only important for your domain range concept but it's going to be equally important when you do application of derivatives with me next year next year almost at the same time you will be studying application of derivatives in mathematics and there you will realize that in the application of derivatives concept there is a concept of increasing decreasing functions or monotonic functions there you will use this wavey curve sign scheme again and that time if somebody says sir what was that scheme i forgot then it'll be very difficult for me to revisit this okay so put your ears like this eyes focus on the screen listen to each and every word which i'm going to say very very carefully okay ready everybody all are attentive cool let's go for it so wavey curve sign scheme as i already told you works under two important situations both the situations should be met number one your left hand side term that you see over here this is this should be in a factorizable format okay and the right side number that you see over here should be a zero then only wavey curve sign scheme which i'm going to talk about in few minutes that will work as it will not work so if you have a non-zero number over here so whatever approach i'm going to tell you that will not work okay so how how will i make use of this method so you have to bring the one to the other side and do some recalculations okay second thing it will not work if you are not able to factorize this rational function because everything is a game of factor over here so wavey curve sign scheme will work only when this rational function is factorizable that is this condition is met and the right side should be a zero this inequality could be anything it doesn't matter whether greater than is equal to whether greater than whether less than whether less than equal to no problem with that the only these two conditions must be there for you to start the process okay so if you are able to figure out that any of the two situations is not met wavey curve is not going to be very helpful okay it's not going to be helpful you have to start looking for you know ways to make it to that form anyways so what is this method all of you please listen to this very carefully this method says step number one make a number line everybody knows how to make a number line okay on that number line write the zeros of every factor that you see in this rational expression whether in the numerator or denominator doesn't make a difference so for every factor that you see there will be a zero for that factor write it on the number line for example this guy x minus one cube the zero of that particular expression will be one everybody understand the meaning of zero of a factor right zero of a factor is that number which will make that factor zero okay so one will be a factor from there right so please put one on the number line so I'll be putting one over here let's say okay next minus two next three okay by the way I have I put one more here so I'll just put it somewhere else because I have to accommodate zero also I have to accommodate minus one also okay so what I've done is on a number line on a number line I have plotted the zeros of each of these factors so one I have plotted see here minus two I've plotted three I have plotted zero I have plotted because this guy is as good as x minus zero to the power six if you want you can write this in your mind like this x minus zero to the power six okay same as x to the power six and of course minus one everything has been plotted okay so first step is clear to everybody step number one is clear to everybody okay step number two is where don't let me not put numbers here because I would not require step number two is we have to assign signs to these intervals that are created as you can see on your screen there are one two three four five six intervals created because of those five numbers right so those five numbers have basically broken down your real number line into six intervals now what I need to do I need to assign signs to those intervals what is that sir what is the meaning of assigning sign to these intervals very simple pay attention what I'm going to say basically what I'm trying to see is that in this interval if I take any number let's say I take this interval okay this interval okay in this interval if I take any number any number no matter how big you want to take what is going to be the sign of this guy in that interval the one which I'm circling out okay so I just need the sign I don't need the value okay so let me pick a number in that interval tell me anything which is more than three anybody any number more than three any number four that's actually this is four so if you put a four in place of x in this entire expression what is the sign of the answer that you get so you'll say sir four minus one cube is positive four plus two square is positive four minus three to the power five is positive four to the power six is positive four plus one to the power eleven is positive so the sign of the number that I will be getting here will be a positive quantity so write a positive sign there okay now rest for sure that if you take any number in that interval so you take you took a four you two you take a seven you take a hundred you take a five million that answer will be a positive you can check yourself you can try to check yourself take any other number take seven those who are saying seven take hundred if you want to take hundred this is the sign sign will be positive for sure convince everybody are you convinced are you convinced everybody yes okay are you convinced what about others are you not convinced convinced okay good now once you have given a sign to this listen to the next step very very interesting step okay start moving towards the left start moving towards the left of this number line okay so start moving towards the left when you're moving left you will see numbers coming your way the very first number that you see is a three isn't it three comes from which factor three comes from which factor in this expression you'll say sir three is coming from this factor x minus three to the power five okay on that factor see what is the parity of the power on it parity means whether it is even or odd okay here you see five is a odd number if the power is odd if the power of that factor is odd then switch the sign from whatever you had on the right to the ultra on the left for example you had a plus then switch the sign it will become a minus so this interval will become a minus got it you can try to test this by choosing any value between one two three let's say i pick up a two so two if i put in this expression you will realize that you'll end up getting a negative answer how let's check if i put a two here in the very first one i'll be getting a positive answer this will be also giving a positive answer but this guy will give me a negative answer this anyways will be positive anyways will be positive so overall this expression will become a negative quality so this is what this negative sign basically shows so these signs are basically showing what will be the sign of that particular rational function in that interval of x and you can take any number between one two three you can check it out you can take one point two you can take one point eight you can take two point six check it out it'll be negative only okay i will tell you the reason for it don't worry little later on first note down the rules now a question will arise and what if that power was even if the power was even then retain the sign very important guys please listen to this very very carefully later on don't ask me sir what used to happen for that case what used to happen for that case i will not be able to tell that time so listen very very carefully if possible when you when you want to see this recording again please do that okay is it fine next next number that you see is a one one comes from which factor you'll say sir one comes from x minus one factor what is the parity of the power on that factor you'll see against an odd number if it is odd then again you will switch the sign from whatever you had to the opposite of it so if you had a minus here you will switch to plus so switching of signs will happen switch means the other one okay whatever you have written to the right you switch the one you switch that number sign and write it in the next one is it clear okay no doubt next number you see is a zero zero comes from this particular factor and this factor has an even power on this even what is the rule retain the sign that means it was a plus so retain a plus sign here clear clear any questions any questions now don't ask me why because i will tell you anyways later on okay next minus one comes from this factor which has again got an odd power on it odd means switch so whatever was assigned here switch it and make it negative so if it was positive make it negative had it been negative then you would have made positive switching switching of the sign and i hope you understand the meaning of the word switch okay next minus two comes from this factor which has got an even power on the top even means retain so minus will remain minus so what do you see on your screen is i have assigned signs to these intervals okay now this can help us answer this question very very easily but before i move on to finally writing the answer to this question is there anybody who has not understood the meaning of odd means switch even means retain odd is switch even is retained odd means switch even is retained anybody has any questions any doubt any questions i hope everybody is attentive i'll take a lot of questions on this huh okay great so if you have understood these two facts then after that is just a you know logical conclusion of this particular problem now you want to write that interval where this particular rational function is greater than or equal to zero so basically write down those intervals where this function is either positive or equal to zero so whenever you have written a positive sign please mention those intervals like this so i have mentioned positive sign here okay you can see my pen hovering around it so that interval minus one to zero please write it down okay note that i'm not putting any brackets around it as of now i will put it eventually but right now i'm not putting any bracket okay next interval is zero to one that also i will write it down and the last interval is three to infinity so what i've done i have written these all intervals one by one any doubt with respect to this any questions any kasta no kasta okay now ensure that while you're doing this you have not missed out any x value for which this expression was even zero so have i missed out one check no one is already written have i missed out minus two yes i have missed out minus two so minus two you write separately over here okay next have i missed out three no three i've already written over here okay so please do this particular thing first of all next time to put brackets around these numbers so what brackets i will put please listen to this very very carefully minus one can i include minus one in my x value look at the expression carefully and tell me can i include minus one so should i put a round bracket or should i put a square bracket what do you think arnav is saying no sir you cannot put a minus one in your x value why because if you put a minus one denominator will become zero and that is like you know making it undefined that is completely unacceptable isn't it so right absolutely round brackets will come here because i don't want minus one to be taken by my x if my x takes minus one right isn't it denominator will become zero expression will become undefined okay zero can i take zero can x take zero tell me can i take zero no because x to the past six will become angry you are making me zero how can it be possible right so escape pass the round brackets okay so zero is here also so put round brackets here also okay one can i put one you will say yes sir so one you put square bracket because one can be taken by x why because at one it will become zero and zero is fine as per my inequality zero is fine can i include three you will say yes again square bracket can i include infinity never sir infinity how will you include infinity okay infinity can never be included okay now what is this single guy doing if there is a single guy you put curly bracket means that will become a singleton set so while these are intervals this guy is a singleton set okay now what do i do with these four sets i have got take their union so your answer is if your x belongs to any one of these or union of these sets it will satisfy the inequality so this is the answer to this question this is the answer to this question okay achindya has a question if two are left then they will become set or they set as two singletons yeah if if two elements are left they will become a set of just two elements so that also will you put under a curly bracket on me correct achindya achindya okay no no no no no no if you write it as minus two to minus infinity first of all you should never write it like this minus infinity should always be towards the left when you are writing an interval guys i have seen this practice wrong practice done by people they will write the larger number to the left and smaller number to the right no smaller number will always be left larger number will always be right so it will always be minus one to zero not zero to minus one okay anyways to answer that question aria you cannot do that because it is not an interval that you are including you are only including one value of x undu value one value not interval aita got it okay so now using the same sign you can also find out where is this particular rational function less than equal to zero less than zero or greater than zero everything can be found out from this diagram okay so even though you're trying to address only this question but this diagram can help you solve solve all types of inequality greater than equal to less than equal to greater than less than everything by looking at the sign diagram is this fine okay now what is the reason for this particular sign see try to understand it there is a hidden region which i will try to explain it in a very simple way see when you are writing i'll just take a simple example let's say i'm writing something like this x plus one x minus one the whole square and let's say x minus three the whole cube divided by x plus one to the power of four let's say i want to solve i want to basically solve this inequality i'm taking a lighter version i don't want to take a bigger version like this so let's try to understand what is the reason for that sign that i was putting okay because many of you would be wondering sir why you are switching the sign for odd and why you are retaining the sign for even i'm trying to answer that question okay now listen to this pay attention uh let me let me take a simpler one here i've taken too high a power let's say i take a simple okay let's say i want to solve this first of all i would like to convert this to a polynomial inequality so how do i do that can i take this guy on the other side and write it like this can i do that can i say this and this are same things can i say that yes no maybe what do you think no i cannot do that okay guys and girls please please please let me tell you this and this are different things see we all have this wrong conception in our mind of course in junior classes we are not told you know much about it when somebody sees this or somebody sees this he basically thinks this is as good as saying a is greater than equal to zero he thinks that this is same as this no you can't just take b on the other side and make multiply it with zero and make it zero see i'll just give you a simple illustration where this particular thing fails we all know that minus one by minus two is half correct half is greater than equal to zero nobody will deny it it's a universal truth okay can i say minus half is greater than equal to zero just by taking this minus two on the other side this will be wrong no minus half is not greater than equal to zero it is lesser than zero correct so we have a very wrong conception with respect to inequalities but don't worry i'll be taking care of inequalities in a separate chapter for you which is called uh inequality theory of equations and inequalities okay that is another chapter which is quite a big one so many of us we have this wrong notion that we can take this quantity to the other side and vanish it no it all depends upon the sign of this number whether it is positive or negative so if your a by b is such that b is positive let me write it here b is positive if b is positive then you can convert this to a greater than equal to zero but if your b is negative the same inequality will convert to a less than equal to zero so it will switch the sign are you getting my point okay so till you know the sign of this number whether it is positive or negative you cannot you cannot take it to the other side because you do not know the fate of the inequality whether it will be same inequality or whether it will switch okay anyways so i cannot do this this step is prohibited okay so what can i do simple as i told you if your b is positive then i can say a is greater than equal to zero so why don't we make our b positive by multiplying and dividing with x plus one itself both in the numerator and denominator okay so if i'm multiplying by the same number in the numerator and denominator the fraction or the rational function is not affected isn't it now i know this guy is positive so you can send it to the other side and you can write it like this so what has happened i have converted it to a quadratic uh sorry i've converted it to a polynomial inequality okay now think as if you have a graph like this okay this is a bi quadratic function right bi quadratic polynomial function you already know bi quadratic polynomial function the end arms will be up in the air like this get it so if you sketch this graph roughly okay now see what will happen this graph is going to cut the x axis or touch the x axis at minus one three and plus one so these are the factors as you can see it will is going to either touch the x axis or cut the x axis at minus one three or one isn't it now all of you please pay attention since the arm is upwards we know the end arms should be upwards this is how the graph should look like it will come down i think i should draw it more vertical up like this yeah yeah so it'll come down it will cut the x axis at this point let me make it yeah it will cut the x axis at this point it will return back it will again cut the x axis at one it will again return back it will come and graze at three and go back again so this is how the graph will be looking like now many people ask me sir how do you know at one it is cutting past minus one also it is cutting past but three it is not touching and going back see it is because when you have a even power on a factor then left and the right of that number the sign is not going to be influenced for example if this number is a three okay if you take 3.1 this expression will still be positive if you take 2.9 that will still be positive are you getting my point so what you draw while you are drawing this sign is actually you are drawing the position of the graph above or below the x axis so if it is above it is positive okay this is also above positive so this is below this is negative this is above it is positive so when when you assign the signs here in this particular number line you are basically trying to make you're trying to estimate whether the graph was above the x axis in that region or below the x axis in that region so above or below will help you to know the sign of y indirectly helping you to know the sign of this particular expression and indirectly helping you to solve this inequality are you getting the point so this is that remote linkage to this concept so it is you know it happens though many times our relatives when they come to see you as they say I am the uncle of your aunties you know this this and that and that is very far off relative okay so this entire concept is basically coming from this polynomial in separate cell okay if you are interested in seeing the graph of this polynomial I will show you because many of you would be would like would like to see how is this touching and cutting happening so everybody please note this down and dictate it to me when I asked you or somebody can take the initiative please note this down I'm going to GeoGibra and I'm going to sketch this out by the way my memory should be able to remember this as far as possible I have absolutely short memory so y is equal to if I was not mistaken it was x minus 1 x minus 3 whole square and x plus 1 right I hope I have correctly copied it correct no no there you go see this is what I was trying to tell you I go small small small small small yeah so as you can see this function is cutting at minus 1 let me remove other things cutting at minus 1 cutting at 1 but touching at 3 right so when you're trying to make a wavy curve for this inequality this is how you're going to react okay so let me make it here so whenever you're trying to solve this inequality what was the approach which I told you make these numbers on the number line okay choose the sign to the right most interval so if I put a 4 I will get a positive quantity now start moving towards the left the first number you see is a 3 3 is subjected to even power so there will be no change in the sign plus will remain plus retention of sign one comes from this factor which has got an odd power on top of it odd means switch then minus 1 comes from this factor which has again odd power on it so that means again you have to switch so this will become a plus so this plus plus minus plus basically shows the way this graph is positioned above or below the x axis whenever it is above it will show plus in that interval whenever it is below it will show minus in that interval okay so I didn't want to go into this depth because anyways I would you know I would have taken a session on this in our theory of equation chapter but this is the reason behind it maybe your school teacher would have also told you why do we take up plus minus like this okay Achin the answer question sir where here why didn't we eliminate the denominator we did not eliminate the denominator we did not eliminate it we basically multiplied with the denominator once more so that it gets a even power on it and if it gets an even power on it we can basically take it to the other side and vanish it off thereby leaving a polynomial inequality Achindya and polynomial is something which we have already learned we know how this graph and all will come so from there we are basically diverting our analysis right so we are basically diverting our analysis to solving a polynomial inequality which is easier for us got the point okay so I hope this process is well as well understood by everybody so we will take few questions because I want you to be very very good with your wavy curve sign scheme okay the other days the senior was your senior was saying sir we can use the baby curve sign scheme it's not a baby curve it's a baby curve okay we'll take few questions okay then we'll see the application part so solve this inequality everybody x plus two to the power of three x minus two to the power of four x plus one to the power of six divided by x minus seven to the power of five x plus four to the power of two and let's say x x x plus three to the power of power of power of one okay solve this inequality okay I hope I have not given you a very heavier one because if you know your basics you should be able to get this also don't be scared no don't get rid of the denominator etc just follow the rule of solving it when I explain the process basically in that I basically got rid of the denominator but while following the actual method don't get rid of anything think whatever was told to you in the left side of the screen that is what you have to follow okay I told the reason for doing it in the right side reason is don't don't you know implement the reason right now implement the actual steps if you're done you can just say done also there's no need to type out your answer but if you can do so nothing like that could you go back to what do you want to take some reference on there okay I'll just go back momentarily and come back again awesome Achintya has given an answer but Achintya will see that meanwhile Arya wants to copy something done done and done okay we'll go back don't worry you will not get such a heavy inequality in your school exams or couple day exams but the concept is not difficult either right if you know how to solve it whether you raise it to power of five or six or a hundred or two thousand doesn't make a difference done Arya is done Achintya is done who else come on guys gently get oh we'll have to take a break also auto is done very good Jyothin is done Shashank is done Aryan J is done very good Vishal is done Tanvi is done okay in another 40 seconds we will start the discussion so those who want to give the response as they're done or the answer both are welcome please do so then we'll start the discussion now after knowing the concept we will see its application also don't worry about it so we'll see it in our domain range concept so how is this helping us in our domain range concept we will see it in sometime okay should we start the discussion okay first thing first thing does it meet the requirement for me to use a wavy curve you will say yes sir left side is completely factorizable right hand side is zero that is something which is very important don't miss out on that don't again jump to the wavy curve blindly first see is it a case where I can apply wavy curve yes I can apply left side I have a factorized rational function it should be factorizable right side I have a zero in between whatever inequality is there I don't care I can solve all of them okay less than less than equal to greater than greater than equal to all the four cases I can solve so first thing is once you have done that please write down the zeros of each of these factors on a number line I can see the least of them is minus four so I'll start with the minus four okay then I think the next one next one next one is a minus three so minus three next one next one is a minus two next one next one next one next one is a minus one then I can see a two and then I can see a seven so 123456 123456 yes so all six factors have me listed down on this number line is this fine everybody I hope you have listed down the correct zeros on the number line okay now start with the right most interval so the right most interval this interval and choose any value in that interval so choose any value more than seven anything let's say I choose a 10 10 plus 3q positive 10 minus 2 4 positive 10 plus 1 to the power 6 positive 10 minus 7 to the power 5 positive 10 plus 4 to the power 2 positive 10 plus 3 to the power 1 positive everything is positive so this guy is bound to be positive so write a big plus in that now start your journey towards the left what is the first number I see seven seven comes from this factor the one which is sitting in the denominator and it has got odd power on it odd means switch even means retain odd means switch even means retain okay so I'm singing like I'm teaching a kindergarten students so seven has this guy has got a odd power so it will be switching so plus will become minus clear is it clear okay next guy is a two two comes from this factor which has got an even power on that even means retain so whatever you wrote here the same sign you will retain over there so minus was there so minus will only come there got it next minus one comes from this factor x plus one which again has an even power so again you will retain got it now minus two comes from x plus two to the power three which has got odd power odd means switch so minus will become plus clear minus three comes from x plus three which is again an odd power again means odd means switch so plus will become minus and finally minus four comes from x plus four square square means even power even means retain so if this is the science you have bought for these intervals then you are on the right track if you have made a mistake in these signs means gone okay because if these signs go wrong then nobody can save you the problem is bound to become wrong the answer is bound to become wrong okay oh see I change the answer no problem I change yeah we are learning it just the second problem that you are doing on this concept never mind even if your mistake has happened not to worry now what are we trying to solve less than or equal to zero means I am trying to see in which interval is my function this rational function whole function either negative or equal to zero so wherever you have written a negative sign write down those intervals so first interval that I would write is minus infinity to minus four then minus four to minus three then minus two to minus one then minus one to two that's it anything that I have missed out okay now wherever it is becoming zero that also should be you know taken apart so zero it becomes at minus two but minus two have already included so I don't need to write it again it becomes a two two is also included I do not need to include it again minus one minus one is only included no need to write again okay so you don't have to write separate numbers okay at least now coming to the brackets around these numbers what kind of bracket will I put thank you thank you two to seven is also there correct sorry I didn't see that two to seven thank you so much okay so negative sign is here here here so five intervals welcome okay thank you Jatin and Shashan thank you so much now I have to put brackets around these numbers so what brackets will I put cf minus infinity is always round brackets infinity minus infinity by default round brackets okay minus four can I include minus four you'll say sir no it is in the denominator minus four factor is x plus four whole square it is sitting in the denominator you can't put minus four because it will create zero in the denominator zero in the denominator will make things undefined so this also has to be round bracket so this also will become round bracket because both are minus four okay minus three again is sitting in the denominator round bracket minus two is sitting on the numerator which can be included so square bracket very very important okay minus one is again sitting on the numerator so again square bracket so this will be square bracket once again okay two will be again sitting on the numerator so again square bracket square bracket seven is sitting in the denominator so round bracket I hope you are able to understand again you are able to understand where I am putting round bracket and square brackets hey seven to infinity will not come to you and seven to infinity it is positive I want less than equal to zero my dear less than means negative why are you including seven to infinity even we are we are basically looking for this interval this inequality you know less than equal to zero less than means negative or zero right so negative that interval you need to copy paste okay okay now we'll have to take the union of our answer there you go and that basically solves the question so your x should belong to this interval done and that's it is it fine how many of you how many of you got this exactly correct I would like to see even if you have not don't worry let me check Achintya is claiming he has got it correct Achintya is claiming he is very good Achintya very good awesome awesome mess up with the brackets most of you are saying brackets went wrong no issues no issues we'll take more questions one more we'll take early before going for a break is it fine okay and then we'll see the application part of it okay I hope you're not hungry stomach is doing good good sir let us have a break good Vishal okay let's take this question solve this inequality 3 minus x to the power of 5 x minus 2 to the power of 4 upon x plus 1 to the power of 3 this time let's solve only this guy less than zero solve this inequality okay Jatin very good Achintya why are your answers different Achintya and Jatin's answer are different okay anyways we'll check we'll check we'll see who is right who is wrong Harshita very good Vishal very good Shoshank very good are you no need to type and all are yeah just say done it's fine it's more than enough for me Arnab is also good okay should you discuss it now okay let's discuss it not a big problem to solve simple ones we just have three factors involved so first of all number one check is it a question where I can apply every curve science scheme yes left hand side is rational function fully factorized right hand side is zero very important okay if right hand side is not zero you have to bring it to the left hand side and again do the simplification before you proceed okay that type of question also we will see in future not immediately okay anyways so what are the factors involved over here the factors involved over is 3 minus x to the power of 5 x minus 2 to the power 4 x plus 1 whole cube so I think the least one of them is minus 1 then I have a 2 then I have a 3 now pay attention here many people have this wrong conception that the right side is always positive so I wanted to break that conception over here okay so many times because in the previous two questions they would have seen that only plus is coming to the right most interval they they take it for granted they think that this will always be a plus no not like that if you not take a number which is more than 3 let's say 4 this guy will be negative this guy will be positive this guy will be positive again so overall this will be negative and I'm sure many of you would have made that mistake taking the right most interval positive by default is a blunder I would say okay don't be so overconfident that you make that mistake is it fine is it okay now start moving to the left side by the way I'll remove these signs before I start the proceedings start moving to the left so first you see a three three comes from the first this guy having odd power on it odd mean switch let me write in white only yeah odd mean switch so minus will become plus then 2 comes from this factor having even power even means retain plus will remain plus then minus 1 comes from this power which is odd odd mean switch so you'll have a minus plus plus minus inner intervals okay I hope people have got this right next what are we trying to solve we are trying to solve those intervals where it is negative less than zero means negative so wherever you have written a negative sign state those intervals okay don't put a bracket right now we'll put the bracket little later on achar bracket also the moment you realize it's pure inequality that means less than only there will be no square bracket ever so by default it will leave everything round brackets so don't waste time round round round round take the union so anybody who got this as the interval of x is absolutely right nothing got it great nothing very good anybody else who got it let me check let me check vishal got it very good vishal who else hashita where is 2 coming from oh you know I said you have included 2 by mistake now 2 will not be included because yeah that that doesn't have to do anything with that is it fine okay not too many guys and girls will be resuming with problems based on the wavey curve sign scheme with respect to domain and range but before that let's take a break let's take a break why is 3 in open brackets everything will be an open bracket sachin everything will be an open bracket because it is not less than equal to 0 I have got it if it isn't equal to 0 then I would have put a square bracket next to 3 okay so right now right now I know my time is hidden yeah right now the time is 621 let's take it 622 we'll meet exactly at 637 p.m after the small kit can break no worries I will not change the site okay see you after the break all right that acts now having discussed the previous methodology which is your wavey curve sign scheme now we'll start applying it where is it used okay of course we will not get direct questions on it but there will be a lot of applications that that particular method has so we'll see those applications in the questions of domain and range so let me take the third case where I said you could have a situation where there is a quadratic in the denominator and there is a linear or a constant on the top okay so I'll pick up such a question find the domain and range for the following function a very simple one I'll begin with maybe let's say 1 by 1 minus x squared okay so you can see here we have a rational function which has got a constant on the top and a quadratic in the denominator so how do you find the domain and range for such a question now allow me to solve one of them you'll get an idea from there and I'll give you multiple subsequent questions based on that okay so let me solve one of them for you and then we'll take up yeah and then we'll take up more questions based on the same concept after this so first then it comes to domain the concept of domain still remains the same it will be all real numbers except those values of x which will make the denominator become zero okay I don't want the denominator to ever become a zero whenever there is a rational function like this so this is very simple on solving this guy you only get 1 and minus 1 from here so if you solve this it means you're trying to say 1 minus x square what is this yeah when you're trying to solve this you get 1 minus x square should not be 0 that means x square should not be 1 that means x should not be plus minus 1 okay so basically your domain will be all real numbers except these two numbers okay your domain could be all real numbers except 1 and minus 1 that is not allowed to be put in this into this particular function any question related to domain anybody no problem with domain okay now comes to range this is something which is important so for finding the range the process is more or less the same okay we will first write down y as 1 minus x square okay and then we'll write x or x square in terms of y so this is how we can do it you may write it like this y minus 1 by y also okay now all of you please pay attention please pay attention if your x belongs to all real numbers except 1 and minus 1 can I say x square will be x square will be all values in the positive real number let me write it like this so this term will be greater than equal to 0 and should it not be 1 and minus 1 yes or no if x is all real numbers x square should be all positive numbers including 0 but should not be 1 and a minus 1 anyways it will not be minus 1 so you should say it should not be 1 correct yes or no correct so can I say can I say this means y minus 1 by y should be greater than equal to 0 and y minus 1 by y should not be 1 but this is anyways true this guy is anyways true because if you do y minus 1 by y equal to 1 you will get a shock of your life giving you minus 1 equal to 0 which is anyways not possible yes or so I need to ensure that y minus y should be greater than equal to 0 now what do you have here you have a wavy curve science scheme to apply which we have already done and that too we have done it to a very you know difficult level this is a very simple one of that kind uh you want me to scroll up where you in this this point you want to discuss domain here I want to see domain okay please don't disturb then let me know once you're done we'll go to the previous problem after we're done with this question Arya is that fine see if x is a real number actually x square will be greater than equal to 0 obvious square of any number which is real is a positive number greater than equal to 0 and if you're not allowing x to become 1 and minus 1 which means x square will not become 1 so if your x square is everything which is greater than 0 and not equal to 1 so should be y minus 1 by y as well correct so y minus 1 by y should be greater than equal to 0 and it should not be 1 now this second guy is always true why because if y minus 1 by y becomes a 1 it is like u saying minus 1 is equal to 0 which is not possible okay now coming to this point this is something which we can easily solve by using our wavy curves so now my wavy curve is basically asking us to solve or helping us to solve this inequality so i'll put the zeros of this factors x minus 1 0 is a 1 y factor is a 0 okay all of you please pay attention now greater than equal to 0 first of all let me put the sign so if i put a number if i take any number greater than 1 and try to put it over here i will get a positive answer from there correct take a 2 2 minus 1 by 2 positive number take a 5 5 minus 1 by 5 positive number okay now each of these factors are having odd powers on them even though it is not written it is very obvious they have odd powers so switching of sign will happen now my question sitter is asking me when is it greater than or equal to 0 that means when is it positive or equal to 0 so it is positive in this interval okay so what kind of brackets will i use let's try to write so minus infinity and infinity you'll always put round brackets minus infinity and infinity you'll always put round brackets 0 can i include 0 you'll say no sir because if you put 0 your denominator will vanish so 0 cannot be included can i include 1 you'll say yes sir so 1 can be included okay so the union of these two the union of these two is what we will be getting as an answer to this question is this fine is this fine any questions so this will become your range of this particular problem now we will verify this we will verify this on our geojibra we'll see does our graph and this result sync with each other so is my domain really all real numbers except 1 and minus 1 is my range really everything from minus infinity to 0 and 1 to infinity let's verify it from our graph okay so first note this down if you have any questions to let me know everybody's happy so let's go to the graph of geojibra so i'll mute this and i will plot y is equal to 1 upon 1 minus x square there you go now watch out this graph very very carefully i'll just zoom in a bit you can see that this function spans all the way from minus infinity to plus infinity only at 1 and minus 1 you can see the graph is having asymptotes that means the graph is not not able to achieve those values so 1 and minus 1 see i've drawn the 1 and minus 1 lines so it becomes asymptotic to that right asymptotic means it's not able to achieve that value it's coming very close and close and close and close and close but never able to touch it that is what is the meaning that it cannot take 1 and minus 1 now see the span of the graph along the y axis it's coming all the way from minus infinity see all the way it is coming from minus infinity till 0 0 is not able to achieve so minus infinity to 0 0 it is not able to achieve and then it starts from one onwards and then it starts from one onwards and goes all the way till goes all the way till plus infinity do you see that so the graph tells you everything the graph tells you domain the graph tells you the range in one shot so whatever we have done is in sync with is in sync with what the graph is also suggesting us are you getting this point so you don't have to always sketch the graph to know your domain and range we can do without it as well is this fine now I'm sure you would like to solve more questions on this concept so let's take more questions everybody please try this one out find the find the domain and range for this function x upon 1 plus x square x upon 1 plus x square everybody try it out if you get stuck I'll be helping you out now instead of a constant I have put a I've put a linear term on the top I want to see how you're able to tackle this take two minutes time and if you're stuck I will be helping you out don't worry see root negative one is a non-real quantity action there remember I am dealing with real valued functions right I cannot put anything which is non-real into the functions neither can I expect anything to be non-real coming out of the function so don't worry I'll correct that answers of your little later on but let's wait for others to respond let's see what right now your answer is correct now your answer is correct see what you wrote was an incorrect statement you are removing a non-real quantity from a real quantity which is said doesn't make sense if your answer is if you're asking whether your answer is right or not to aria I would say no that is wrong we'll discuss it aria no issues try it out answer is correct harsita domain is correct go for range range is slightly challenging as you will realize in some time that finding range is not that easy as finding domain is that's why people lose marks not in the domain questions but in the range questions okay a chintya good can you write that as an intro a chintya can you write that terms in terms of an intro absolutely correct good again minus one should be to the left remember in any interval that you write whether open interval close interval the smaller number should be on the left the larger number should be on the right very good a chintya anybody else who would like to contribute to the range part of it I think domain many of you have given the right answer or even if you're stuck tell me I'm stuck so that you know I know how many of you are stuck somewhere we'll discuss not necessarily not necessarily aria you can do without that as well okay I'll give one more minute to the people who are trying hard and then we can start the discussion yeah so if it is not possible then don't remove anything no don't remove anything basically what does the definition say remove those values of x or those real values for which the denominator becomes zero and if you think denominator doesn't become a zero then don't remove anything so your answer becomes all real numbers right jethin yes shall we will discuss it guys thank you all of you have given a best shot you have tried hard let's discuss it so for the domain the process is the normal one which we have been following since the beginning of the rational function discussion all real numbers except those values which will make the denominator become zero but there is no such value of x which will make denominator zero so this is going to be a null set so basically you have r minus null set which is r only so domain is all real numbers is it fine clear aria right so don't forcibly try to make your denominator zero if it is not zero then except the fact that you are not going to remove anything from the real number so your answer is real numbers okay now coming to the range part of it for the range part the process becomes slightly deviated from whatever we have done because in this case you realize you will not be able to less cross multiply you will not be able to separate out the you know x completely in terms of y but you don't have to do that always in this case you will see here that you are getting a quadratic in x so treat this as if you have written a quadratic in x what do you have written a quadratic in x something like a x square b x plus c right so a is your y b is minus 1 c is y something like that okay so when you use the she either our chariot formula or the quadratic equation formula to solve it you get x as minus b plus minus under root b square minus 4 a c by 2 a now you can see i have written x completely in terms of y so let me just simplify this is everybody happy till the step then only i will move forward are you all happy with this so first what did i do equate it to y like the normal way i you know take that took the denominator i got a quadratic in x i use my shithera chariot formula the quadratic equation formula what you know it as and i got my x in terms of y everybody's happy till the stage then only i'll proceed further then only i'll proceed happy okay now all of you please pay attention since your x is all real numbers this is what is our domain requirement it means this guy should be taking all real values yes or no since x is all real number this guy should also be real number yes or no correct now for this to be real number tell me what are the situations or what are the restrictions you need to honor if you want this guy to be real number what are the things you need to honor you say sir the thing which is under the root should be paused greater than equal to 0 agreed next next is this the only condition you need to honor achitya right and somebody would say sir 2 y should not be 0 right guys this is something which we need to actually verify this condition the second condition may not be honored i know you will be surprised right but i'll show you in some situations we need not honor the second condition but nevertheless we will take a cognizance of that and we'll try to verify it whether actually why should not be 0 or why can be 0 also that means 0 chalega correct so achitya and our love point taken but we need to verify the second thing okay let's see we'll do it one by one so the very first guy says this now this is as good as your wavy curve science scheme so i can write it like this by the way all of you know that i can factorize it like this see wavy curve will only work on factors isn't it so i can write it as 1 minus 2 y 1 plus 2 y greater than equal to 0 so for this you know the approach the approach is we make a number line i write the zeros of these factors so i think minus half and half will be the zeros take a number more than half let's say i take a one i'll get a negative so this will be negative then switching off sign will happen two times because they are all factors which have odd powers on them correct now my question sitter is asking me where is it greater than equal to 0 so greater than equal to 0 means positive or 0 that is only between minus half to half okay so your y should belong to minus half to half this result comes out from the first one okay let me call it as or let me call it as condition number a okay condition number b says sorry condition number 2 says why should not be 0 now is it really true is it really true that y cannot be 0 i don't think so because if you go to this expression it really tells you that when x is 0 y is 0 right so what does this say y can be 0 why not why not why can be 0 who is stopping y from becoming 0 nobody can when x is 0 y is 0 so when i wrote a verify here basically it came out that this particular restriction is undue you don't require this restriction so i'll score it off this is a condition is not required sometimes it may be required but you need to verify whether that condition is actually legitimate or not now many people ask me sir why it is working because in this case as i understood that when x is 0 y is 0 but why is it working now all of you will realize that when y is 0 the numerator also can become a 0 if you take a minus sign see 1 minus under root 1 which is 0 0 by 0 0 by 0 is an indeterminate form indeterminate can have a finite answers to it are you getting my point and because of that 0 can be allowed in this case yes i agree to the fact that in some cases you may have to exclude the values for which the denominator is becoming 0 but that is only when you have verified it it's very important even in school teachers make a mistake like this they'll say oh 0 is not allowed i don't first check it out because there is nothing better than this equation to tell that y can become 0 like who is this method to tell me when i know that x is 0 y is 0 finish right why should i exclude 0 unnecessarily why should i exclude you are you getting my point so don't buy these statements blindly that you know since y is coming in the denominator y cannot be 0 okay so the final verdict that you get is only condition number a and this becomes your range as well so range is all values from minus half to half this is your answer simple as that don't exclude 0 you will get a wrong result okay let's do a verification of this from our graph what does the graph have to say let's go to the graph so y is equal to y is equal to x divided by 1 plus x square there you go and you can see this graph domain is all real numbers see the span along the x axis i hope you can see my camera minus infinity to infinity right but along the y axis you can see it is only sandwiched between half and minus half half and minus half and you can clearly see 0 is taken by the answer no see here it is taking a 0 y is becoming 0 here it is passing through that point okay so the second condition that you are getting that needs to be verified please note this down because people have lost marks because of you know writing a blind answer okay is it clear any questions any concerns with respect to the solution of this question can i give you another one will you be trying that out okay achyantya has a question sir in a condition where 0 can't be taken how do we do it 0 can't be taken out achyantya if you if you realize that you are getting a term in the denominator and you are trying to see for what value of y is that term becoming 0 let's say a number will come out from it in this case the number is 0 let's say you had a 2 y minus 1 let's say 2 times y minus 1 then y will become 1 so just put a 1 here and try to solve the quadratic are you getting any real root for that if yes then y equal to 1 is a possibility that means range will have 1 in that in its you know range will have 1 in it but if you are getting a non real answer from there that means this quadratic whatever you got was a non real one that means this equation was correct that y cannot be 1 is it clear yeah so basically what i was trying to say aria was that when you are getting a indication that y cannot take a certain value because the denominator will go 0 try to put that value manually here and solve for x are you getting an x value for which y is taking that answer if you're getting an answer that means the second condition is null and void you don't have to honor it you have to just honor the first time all right let's take more questions i think with more questions clarity will come let's try this one out find the domain and the range of find the domain and range of now see i'm complicating it now i'm taking a question where you have quadratic in the numerator also and quadratic in the denominator also approach is not going to change same approach i want to see whether you are able to take that approach try this out find domain and the range of this so here we have a case where my rational function is made up of quadratic both in the numerator and denominator both are quadratic correct aria very good correct acintia correct jatin very good enough that's right i'm more interested in range let me see who gives me the range answer i'm more interested in seeing that result what do you see blessings from the previous question give me some blessings i'll i'm trying the new one yeah the answer for the range is not matching with yours what if it is not quadratic then fine if it is not quadratic you write x in terms of y you know acintia that's absolutely correct well done boy awesome good absolutely correct i'm so happy to see somebody getting this this right by the way aria can i go back to the the next slide anything that you would like to do okay thank you awesome acintia very good good who else acintia and are you are you related to aniruddha n no i think aria shree aria shree is related friend okay yes anybody should we discuss it partly correct and enough one of your figures is correct but not completely correct never mind let's discuss it out okay if you have any issues i think after i solve this things will be resolved we will check aria we'll check we'll check see so first of all the domain is the easiest part okay so domain is basically all real numbers except those values of x which will make the denominator polynomial go zero but if you see denominator polynomial can never be zero because this can actually be written as x plus half the whole square plus three three by four so this will always be more than three by four it can never be zero and even if you try to solve for it you will get non real values correct correct this will not have real a discriminant would be negative so as to say okay that's another way of looking at it so ultimately this will give you no answer that means r minus null set which means r so domain will be all real numbers that most of you got it right okay so everybody is getting domain as all real numbers so this guy will work for any input you give it to it zero down low no fragmented okay now let's talk about range for range some of you have got stuck somewhere so let's try to figure it out so the range process is same you will have to either write x in terms of y or get a quadratic in x that is the two possibilities that will happen so let's do that let's see which of the possibility arises so if you take the denominator to the right side so left side you'll end up getting x square y plus x y plus y is equal to x square minus x plus one if you take x squares together you'll end up getting something like this if you take x together you get something like this and if you take the non x terms together you end up getting something like this right so treat this as if you have written x square into a or let me write a in a short away i mean in smalls not in gaps so a x square plus bx plus c kind of a scenario just compare it with this so basically what i'm trying to say is that this is a quadratic in x now aria the reason why you lost an x is because you cancel this out i don't know why why on this planet no sir i didn't cancel it out i took uh wait wait i in the first two terms i multiplied it uh x square into y okay i took uh wait something had to come out one second something okay x square i took come out no i got it on the other side okay that x square why will you lose an x correct sir and then from the first two terms i took x common out okay i did something else actually not not lose x square you not lose any of the terms don't worry okay now once you've got a quadratic in x you can use your sridhar acharya formula or sridhar acharya formula minus b plus minus b square minus four ac by two a okay now having got this now please recall what was your domain domain was your x could be any real number which means this whole thing like this guy whole thingy i will write it down again unfortunately okay by the way i'll do some simplification in between between over here so this i will write it as minus two y minus one the whole square i hope you are fine with that because four y minus one whole square i can write it like this okay by twice or five minus one even this should be real yes or no now what are the restrictions we need to entertain or what are the conditions which we need to entertain for ensuring that this is real so you'll say sir number one the quantity which you have written under the root symbol that should be greater than equal to zero agreed why because if this is less than zero under root of a quantity less than zero will yield a negative will yield a non real answer so this whole thing will become non real which i don't want i want it to be real right and second condition you will say sir this guy in the denominator that should not be zero but again this is subject to verification so i will write verification or verify as of now here we will verify it separately whether really why cannot become a one or not we will see we'll check okay first we will try to see what comes from the very first scenario so this guy if you see this on simplification this will give you y plus one plus two y minus one i'm using a square minus b square factorization and y plus one minus two y minus one okay this one simplification gives you three y minus one correct me if i'm wrong and this gives me this gives me three minus y okay so for this let me recall my wavy curve sign scheme that is going to be a very helpful method to solve this inequality so first write down the zeros of these two factors i think this guy will give me one third this guy will give me a three and let's start assigning signs to these three intervals take a number more than three in your mind let's say four so i think 12 minus one is positive but three minus four is negative so this guy is negative and now all these powers are all these powers are all that means keep switching okay plus minus now the question center is asking where is it greater than or equal to zero that means where is it positive or zero so it is clear that it is positive between one third to third and at one third and third it is zero so this is what you are going to get out of the first condition okay so this is the outcome from your first condition let's call it as a as of now okay now the second condition says y cannot be one but i will not take this on the phase value i will go and check whether y really cannot be one but i can get my answer pretty straight forward if your x is zero y will be one so there is no reason to exclude one from your range isn't it so when x is zero you can see yourself that your right hand side will become a one isn't it x is x equal to zero is the real number i'm feeding this as a zero inside it it is giving me the answer as one so why should i exclude one this this condition is absolutely redundant i will not honor this condition that means i will not honor the fact that y can y cannot be one y can be one y can be one very much so this is the only thing that you're going to get so this becomes your answer is this fine i think only achintia got this absolutely correct rest people start and made some mistakes here and there chili mistake chili there is nothing called chili shake chili hi if if let's say one was not possible then you could have written it like that also achintia or you just say minus curly brackets one that is also fine is it okay should we do one more question before we end this session okay by the way there are many more things to be covered up in the next class you'll be talking about rational function sorry irrational functions logarithmic functions exponential functions bit of trig inverse trig and piecewise that should be fast i would not spend too much time on that okay let's take this question find the domain and range for satyajit i i is an imaginary number you're you're trying to say i'm excluding i from real numbers i is anyways not a part of it it's time to say sir i'm throwing you out of my house i'm anyways not at your house okay no sir you're most welcome to my house sir why should i throw you out of my house right aria that's correct in your school primarily they're going to revolve around this concept maybe some some questions on irrational domain and range will be asked which i will take in the next class next class will be slightly faster we will not take a lot of examples but we'll just keep on understanding the concept because i need to start with trigonometry also because it's a huge chapter trigger yeah trigger is a huge chapter and i'm sure you would have started trig in your school trigonometry has got so many subchapters under it okay jatin jatin are you sure because we'll see we'll see give you a best shot everybody we'll see yeah please have a relook at whatever you have done i'm sure something has gone wrong okay vihan guys one more minute i'm going to give you and then i'm going to start the discussion fine okay ashita please everybody participate i should not be taking names swithi shri vankat omesh harsha da avnish see whatever i'm teaching i have not even started with the j part it's just school level okay see our approach is always from school and then higher right so this is all important part of your school syllabus also i hear this lady was blessing on the previous question yes aria tell me where you want to see what do you want to see which wave this guy where we go all right let's discuss it time is less just four minutes two two minute is too much the question lifespan is just four minutes after that it is like either you solve it or you quit we don't have much time i think three minutes is left see it's not a very difficult question just follow it domain will be what domain will be all real numbers except those values of x which will make the denominator go zero but denominator can never become zero so basically it's all real numbers minus a null set which is all real numbers okay so this is going to be a real this is going to be a domain okay all right we'll we'll check the range so for range okay satyajit we'll discuss it aria let's let's figure it out so first of all write it as a y cross multiply i mean take the denominator to the left and write it as a quadratic something like this okay and you can do this as well you can do this as well not an issue okay okay so here it is basically a simpler case because you can write x square completely in terms of y like this correct me if i'm wrong okay now if you know your x is real it means your x square should be greater than equal to zero correct yes or no which means one plus y by one minus y should be greater than equal to zero yes or no this is a simple simple wavy curve question correct so now plot the zeros of these two factors over here so one is minus one other is one let us assign sign to these intervals so if you take something greater than one let's say two this will become three by minus one so this will be negative this will be positive this will be negative again clear now greater than equal to zero means positive or equal to zero so can i say minus one to one but i cannot include one i can only include a minus one because if i include a one y will become a zero sorry denominator will become a zero and if denominator becomes a zero that is not permissible okay so those who have said square bracket minus one to one let me tell you my dear students unfortunately that answer will be sitting in your options and you will confidently mark it right and lose one mark okay be careful see you anyways you can test that this guy cannot become a one if you put this equal to one it means you are trying to say see you're trying to say if this can become a one that means you're trying to say x square minus one can become x square plus one means you're trying to say minus one is equal to one right not possible okay so it can never achieve one it will be round brackets there is it clear is it clear any questions any concerns you got square brackets because of that okay send me the your solution achenya yeah okay so with this we'll stop today's session our next class will be the very vital one because i'm going to go beyond your school level uh concept of domain and range please do not miss out on that class don't be late because i'm going to talk things which may not be present directly in your textbook okay so till then take care good night shabat hair stay safe everybody thank you