 I am interested in, I can differentiate between relations and functions. So I have a question for all of you, if you have understood this concept, well, what is this graph of the function? Of course, functions are relations only. What is function? Functions are special relations. So, will it be a relation or can it be qualified to part of function as well? That is what my question is. It is not a function. Why is it not x equal to a to x equal to b? Let us say my domain is, by the way, I will tell you the meaning of these brackets as well. So, many things we have to learn because we have not been introduced to these small, small things in our class 10. Let us say a and b is your domain. Why vertical line? Why not horizontal line? For the same x value, there are multiple y values, which value is the second criteria. Absolutely brilliant. Bang on target. What he said, I will summarize you in simple language, he said that if I draw a vertical line, if I draw a vertical line like this, if I draw a vertical line like this, for a particular value of x, let us say x1, I am getting multiple outputs, I am getting multiple answers, y1, y2, y3. That means if I draw an arrow, there is an element x1, which is mapping simultaneous, this is not allowed in a function. That means you are putting a two-print responses from the machine on three different days. Of course, you will throw out the Dabba machine. So, this will not be a function anymore. It cannot be a graph of a function. Yes, it is a graph of a relation, but that relation is not eligible to be called a function. So, many such things will be learning in this particular course. So, I will start with the graphs of standard function. So, function should be treated like machine in your mind, where you put in some input and it gives you some output. So, function is like a machine, where you put in some input. You can call this as an input and you get an output for it. This is an output. And of course, now you know that what makes that machine actually a machine or actually a functional machine. Now, I will be telling you the graphs of some standard functions that you will be using throughout your maths. And let me tell you, functions are the basics of all the vertical of maths that you are going to study, whether it is a calculus, whether it is a trigonometry, whether it is a algebra. You are going to deal with functions, Jn and dh. You must also be doing computer science, correct? When you write methods in Java, these are all functions, correct? Everything about some functions, where you have given some inputs, which we call as the arguments. And it returns some value to you, which is called the output. So, this is your output, whatever you feed is an input. Now, you may ask, sir, are functions only meant to take one input? The answer to that is no. There are multivariate functions as well, but that is beyond our scope right now. Okay? Of course, when we do higher engineering, we will come across multivariate functions. I think in electrostatics, when Felix will be talking about potential, potential depends upon the x, y, z coordinates of a body, right? So, there you will be dealing with multivariate functions. But as of now, to keep things very light for you, since you are just beginning to learn all these things, we will only talk about single variable functions. Single variable function means you put one input into that only, can't put multiple inputs at the same time. Okay? So, when I talk about standard functions, the very first thing that comes to my mind is a line. Line is also a case of a function, right? It's a linear function, which we call it. Okay? So, when I say y is equal to mx plus c, how many of you have come across this expression before? y equal to mx plus c. This is called the slow intercept form of the equation of a line. Who doesn't know this? Everybody knows? What is m here? Slow. Slow. What does slope represent? How steep is the line, right? Can slope be a negative quantity also? When is the negative quantity? Yeah. Yes, sir. How is slope defined first of all? Slope is defined as a ratio of rise to run. What is rise to run? So, if I move from this, this would be called my... So, this ratio... Now, where is the slope of it? I mean, how much you're descending as that, right? But if I go... Okay. You're saying if I go from B to A, the same line will... And if I move from... Is that what shares this thing? Is it correct? It depends on what you're taking in your water relay. I'm going to be different if I go from B to A. He'll be negative of that. Many of them disagree with you as I also disagree on that. See, when you say rise are if both are negative, M will be positive for sure. Correct? So, when you're going in this direction of Y, anything measured in this direction of Y is positive, anything measured in this direction is negative, you know that? Anything measured in this direction is a positive X, this direction is... Rise is positive, run is also positive. So positive by positive is a positive quantity. If I went from B to A, rise is negative, run is also negative, so again the ratio is positive. So this line will have a positive slope irrespective of which point to which point you are targeting. Is that right? That doesn't depend upon the two points you have chosen. But however, if I have a line like this, and this is my point, let's say A and this is my point, let's say B. So when I go from A to B, when I go from A to B, is my rise positive? Run is negative. Run is negative. So this ratio will be negative or not? Yes or no? If I go from that ratio will be negative. So such is an example where this line segment or whatever line that is a part of would have a negative slope. Understood? So in simple language, anything which is like this is a positive slope. I am showing it from your angle. Anything which is like this will have a negative slope. Understood? Correct? So when I say this is a positive slope, if it is lesser, lesser, lesser, the value will decrease. If it is like this, the value will increase and it can also become infinity and it becomes parallel to the y-axis. Okay. What is C by the way? Y intercept. What is Y intercept? It's the distance from the origin where the line cuts the y-axis, but this is our directed distance. What is a directed distance? Just like vectors. It also has a positive and negative attach to it. So if it is cutting the y-axis above the origin, C value is positive. For example, if I draw a line like this and I say this line is passing through 0, minus 3, what is C for this line? It's minus 3, don't say plus 3. If I draw a line like this and I say it is passing through 0, 2, what is C here? Plus 2. Is that correct? So everybody sitting here can draw the equation, can draw the, which is a line. If I give you the m and c value, we will be able to plot it roughly. Yes or no? Just a small exercise we will do on that. Plot, Y is equal to negative 2 x plus 5. I know all of you know that, but just for, knowing that everybody is on the same page, just draw this. I don't want you to use KL and all those accurate measurements. Just draw a line, name it at a proper point where I know that it is passing through this one. Are you sure? You are guessing it as well. Where do we get the Y axis? Mark that point as well. Fine. Well, just for your mark, you will take good. So it will be basically a line like this. Where it will cut the Y axis at 0,5. X axis will be cut at 5 by 2 comma 0. Please make arrows to signify that this line is extending indefinitely in all the directions. Yes. Such a relation of a line or such a relation representing a line, is it eligible to be called a function? Correct? Yes or no? Second question is, what do you think is the domain of this function? We talked about domain range, right? What is the domain of this function? X is equal to R. X is an element of R. Did anybody understand what he said? All real numbers. All real numbers. So for him, he said that the domain of this function is all real numbers. What in the graph tells you the domain of the functions? The span of the function along X axis. Are you getting this point? From where to where my function is extending along the X axis, that determines the domain of the function. Understood? So here, I can say domain of this function. I may write Y, sometimes I may write F of X. Don't get confused. Last year, what happened? Sometimes you write F of X equal to this. Sometimes you write Y equal to this. There's no difference. It's because we've got F of X along Y axis, we call it Y also. So domain of the function, as Sher's rightly pointed out would be all real numbers. Now many people say, why do you write this R with a double stroke? Why not a single stroke? Because sooner you realize when you do the chapter relations, R is also used for writing relations. Relations. Just now we talked about relations. So in order to distinguish between two R's I'll put a double stroke. In fact, in books also they would do the same. Just like different people are in the same way. Does this make sense? Does it make sense? If you say R with double stroke, I would take it as a set of all real numbers. But if you just write R like this, I would say, is it a set of relations? An element of doubt may come into your mind. Okay. Anyways. Is this right? Yes. Now, very good question. If I ask you, plot this line, what does this say actually? It says that, I don't care about Y, but X will always be 5. That's a very crude way of saying, I don't care about Y, it's a vertical line. And in that case, Shishti, Anjali, this will not be a function. Why? Because for 5, you are plotting to 1 or so, whatever Y you choose, it is all mapped to 5. I'll get you this point. So it will not be qualified to coordinate functions. So all lines need not be functions. Very valid point brought up by Anjali. This may be a function, but not in this case. But if I say Y is equal to 5, how do you plot the graph? Sorry. Whatever is your X, my Y will be 5. So 1 comma 5, 2 comma 5, 100 comma 5, 0.0025 comma 5. Whatever you choose, the Y coordinate will be 5. Will this be a function? Yes. Why not? Now here comes another misunderstanding hypothetically I draw a relation for you, which maps set A to set B. Set A is for the pre-may set, set B is for the may set I already indicated for you. Let's say, hypothetically this has got elements A, B, C. And this has elements 1, 2, 3, 4, 5. Okay. So consider these questions very, very carefully. Under this relation, let's say A maps to 1, B maps to 3. Is this relation a function? Yeah. C is not here. Sorry. What do you think? Pratam. Yes, Pratam. Is this the function? C is not getting mapped. So how can you call this as a function? It is not. So C is not getting mapped, so it is not a function. It may be a relation, but it is not a function. Chalo, I map C also, but now C is going and mapping to 3. Okay, that means the rest of you say it is not a function? Or how many of the rest say I don't know actually? How many of you say I'm not sure? Not sure, Ashish. Ashish must be sure what exactly you want to write. I'm just kidding. It's actually a function. Now many of you would think that it is not meeting the second criteria. See, second criteria says this fellow cannot go to 2. But 2 of them cannot go to 2. Bread number 1, you put, you get an output. Bread number 2, you get a symbol looking. Let's say bread of one company, bread of another company. So just to show you the graph, one to many, just to make you understand, there is a one here and there are many. Many to one is a function or can be called a function. Understood? So when I draw a line like this for you, y equal to 5 line. Let's say y is equal to 5 line. So if I draw an arrow diagram for it, 1, 2, 3, 4, 5, whatever you may write in. I write numbers from 1 to 5 only. And here there is a 5. Everybody is mapping to 5. So many to one, many to one, this is a function. Understood? So it is not violating that unique characteristic. Unique characteristic is, means one input should give you one output only. Now that output will match with the output of some other input at its time. Understood the difference? Understood the difference? So a practical example apart from this will be a function like x square. Is this a function in real domain and real co-domain? It's a very important thing I wanted to tell you. Whenever you mention a function, don't mention it just by stating what is the relation between x and y. Also mention simultaneously where is the function defined. And the best way to write this function would be something like this. What does this small statement signify? It signifies that this is a function you may write y, you may write f of x as your call. Don't get confused between y and f of x. They are one and the same thing. Okay so this is the information about the domain. Remember this? When you mention this first set is actually telling you what are the permissible inputs you can put to the function. It's like when I sell a machine to you, I give a user manual along with it. So when I sell a toasted to you, I should be put in this toasted. Don't start putting your books and shoes and school bag inside it even though you hate them. So that user manual as to what you should put in that machine is what? This. Beyond this you cannot put anything. Remember that. When you buy a car, what is the car? Only to be driven on it. You can't take it in the water. What is this range? No that is core domain. What outputs you can expect from it? What outputs you can expect from that machine? But it doesn't tell you what outputs you can actually get from that machine. Understood the difference? So I gave you an example of the difference between core domain and range. One, two, three, four, five, six, I wrote that then right? And I told you core domain is an entire set. And range is the values which are actually getting mapped. I give you a simple example. This pool is like the core domain set. You can go to any room you want. You can. But do you actually go to any room you want? No, you actually end up going to your class lab washroom. So class lab washroom will constitute your range. But you have been given to access to go anywhere in this room. And you can also enter the principal's office. Of course without permission. But that is not somewhere where you would like to go. Unless until you have been called. Yes? So the entire school is your core domain. But the places which you visit is your range. So this tells you where all you are allowed to go. Range is where all you choose to go. Understood? Understood the difference now? Whenever you write a function, always write this along with it. Never try to sell a machine without the usual money. Simple as that. You tell the person that you are using this function. Only put these values. And expect this to come out. Understood? Now, is this function eligible to be called a function? If I put an input to what I would know. Even though it is fine that we are not a function. Understood though? Will not make a mistake about it? Now many people ask me. Is it fine to leave an element of the core domain set unmapped? Yes. But it is not fine to leave an element of the domain set unmapped. If any one of them is left out in this set. Then that will be not a function. It is fine to leave out elements there. Understood now? So 1, 2, 3, 4, 5 will be your core domain. 1, 3 will be your range. A, B, C will be your domain. Is that right? Is that right? Of a state line is clear to everyone.