Hello students welcome back Gajwakr academy my self Randeep Singh and we are discussing class 11th applied mathematics new chapter number nine functions so last we discussed chapter eight now new chapter has come to you chapter number nine functions so whoever knows this chapter is similar to the NCERT book that they have, that means those who are studying science maths using their NCERT, this chapter number I think is the second number chapter there you have the first and second chapter, that chapter is functions okay it will either be first or second for you which book do you have M L Aggarwal in this book this chapter number nine means functions okay see the function chapter is very easy you just have to learn a little bit of language in it there are some terms you will have to understand those terms if you know those terms well Once it is clear, you can use them to generate questions. Okay, so I will explain it to you a little bit here in this introduction part. In exercise 9.1, we will discuss all the terms that can be used, all the concepts that can be used. And the most important thing here is the theory. So I will explain to you once, all the theories written in the book, okay. The biggest doubt is regarding the theory in this function chapter, that the students are not able to relate the theory with the questions. So I will keep giving you the theory with examples, okay, so that you understand what we actually call a function. It is clear, let's start without delay. And all the students who are new have already started the lecture, so please subscribe to the channel, okay, and like and share it. You guys must wait for the chapter to end. After this lecture ends, let's start with a function. Now see what a function is. A function is a special case of a relation. Before that, let me tell you what a relation is. Relation means there is a relation between one and the other. For example, I have written it in front of you here x = y s x = y s. As soon as I write the value of y here, you know that you make coordinates. You have studied geometry in class 10th, so we used to make relations in geometry, how if I write the value of y as one here, then what should be the value of x? It should be one. If I write the value of y as 0, then what should be the value of x? Four. How, like here, if I am putting two, then the square of 2 means 4, so the value of x becomes 4. How do we always write a pair? How do we write x y? You just have to understand this pair. This concept is there in this entire chapter, firstly in exercise 9.1. Okay, so the function is a special case of a relation. How to specific let equals two non-MT sets. I have created two sets, right? This is an example. I am explaining the theory to you by giving direct examples, so you will not be able to relate the theory to whatever is written in your book, so try and relate some examples from your theory with my concept, then you will see how wonderful the concept will develop in you, we have taken two non- MT sets, let's assume one to two in this, let's assume two and three in this or let's assume four and three, set and r and f b relation from x to va, now I have created two relations here, I am naming it x, I am naming it va, so if we do not call these x, we would have called these x and va, now this is creating a relation of x to x va, then r may not relate an element of x to an element of va and it may relate an element of x to more than one element of va, what does this mean, it could be that all the elements in x and va are related to y, okay, like 1 is done with 4 and suppose 2 is done with 3 or suppose one is done with four and one more element is given from one, here five is done but one element is left, that is what I mean to say is that in the relation r, there may not be an element related to x and y, that means complete elements of x to y are related and it is possible that complete elements of x to y are not related, now the relation which you are telling that we relate, how will we relate, I will tell you how will we relate, okay the concept is, but a function relates each element, now when the function says this, brother, we know that we can make the relation in any way, so how will we make it, but what does the function say, the function relates each element of x, whichever element of x is to a unique element of y, that means it should have a unique element, like suppose I have written the values ​​of one here in the function And then we write 5 6 for va, so each element of x, every element of x, must have a unique element, y must have a unique element of y, like 2 has 5, r has six, four has nine, okay, for example, if this relation of 4 was not with 9, let's say this would not have happened, and r would have been with nan, and here a 10 would have been written, now see, does this 3 have a unique element, no, it has two elements, there is no unique element here, did you understand the meaning of unique element, only a single one, what do we call this, we call it image, now I will explain further, you have terms for all these, what do we call this, we call it image, so what does the function say that the function relates that each element of x must have a unique element with va, okay, okay, now a definition of this comes out, what is it, starting from this concept Look at the definition, if xva are two non mt sets then subset of va subset f subset we have created a function subset now you will ask sir how this subset has been created, suppose we have multiplied xva and va, we have multiplied xva, like I am giving an example here there is two in this set in x set and va has 3 4, if I multiply it then what will happen I am telling you first one will be multiplied with th, then one will be multiplied with four, then two will be multiplied with three, then two will be multiplied with four, now I am creating a subset here, I am taking a subset of it, subset means I will make any of my pairs from this and will make such pairs which should be functions, I will make such pairs which should be functions, okay suppose I have taken a pair from this, one and three and two and four, what is my subset, there is one from this, we call it a function, mapping It is said and map is said from x to y. Now keep in mind that the function we are choosing for x and y, which is a subset, in this every element which is in front x should belong to the set of this x and the element which is behind y should belong to the set of the previous y, did you understand, here it is written x belongs to x there exists a unique y belongs to y, that is x y belongs to f, tell me, did you understand, what is this f, your same function which is a subset of x * y, tell me, did you understand, okay, now this function which has been formed for x to y, it is necessary to have two conditions, you need to have two conditions, which are, now the subset of x y is called a function from x to y, okay, we call it a function of subsets, okay, now what is its condition, if for each x the next value of x belongs to the set of x, there exists y, this It belongs to the y set, it is true that now the one bracket that is being formed by you belongs to the f function, okay, first thing, second thing, no two different orders have the same first component, now what does this mean, it means if I have written one and three, now if suppose I write one and four, then it will be wrong, why, because these are those orders, these are those orders which have the same first component, what has happened, the same first component has come, now this means let me take you back, see, this is the same thing, three is getting two answers, three is getting two answers, see here, three is also coming near one in the answer and four is also coming inside one in the answer, it means when we will do its diagram, that is, its mapping, then one is also related to three, four is also related, this does not happen inside the function, tell me did you understand, you can choose only those order pairs from this, in which the first is there. The first component of the component is not coming out the same, tell me did you understand, is it clear, what am I trying to say to you, that is why I have used the word, in the function f we use subset, inside x * y of x * y, the same components will also come, the first component which is same and the different ones will also come, we can only use the different ones in the function, tell me did you understand, is it clear, did everyone understand these two conditions, brother, tell me, tell me quickly, I am telling you very well, you have to understand it very calmly, there is no need to hurry, okay, you just have to remember these two conditions, okay, there should not be the same component, same means they have different, here the same number has two different images, the concept of unique images among themselves has been cancelled, okay, we do not call that condition a function, these two conditions should be there, then we will call it a function, it is clear, let's move ahead here, next see in other words A function from x to y is a rule of correspondence what does correspondence mean first of all I understood its second meaning something different which associates to each element of x of x a unique element of y to y right you have to alter it completely now the next term used here is image now what is image oh friend just now I told you this y which we are finding the answer to these y these are the values ​​of y which belong to the y set what do we call these we call them image what do we call them image so image of an element the unique element y belongs to y is called the image of an element x of x under the function x2y tell me did you understand is it clear here everyone let 's move ahead ok let's see further it is denoted by fx5 y = fxxx.pro what can we call function isva we can also call it value of function at x we can call it x over x okay Let me give you a wonderful example. y is an x ​​condition. If I put the value of x at one, if this is what it is, I can also write it as f, a function. If I put the value of x at one, then what will be my answer? By putting the value of x at one, what will be the square of one? If I put the value of x at 0, then my answer will be the square of t, that is, four. That is, I am looking for the element y that is coming out. What is this? This is the value of the function. The value of the function is 8. What is the value of x? Here it is 2. 8. But tell me, did you understand? Is it clear? Look here, these are the terms, it is very important for you to understand them. Until these terms are not clear to you, you will not be able to understand the concept of function in depth. That is why most of the students have doubts that Sir, they get into the condition of rote learning the concept of function. They will start memorizing the questions and I am telling you, please read it once according to me. Seriously, you should finish one lecture, repeat it once, understand the basic concept, after that solve the example, solve the exercise, you will become so brilliant in the concept that after doing it once or twice you will achieve perfect perfection in the exercise, you should assume that you understand what I am trying to say, so this is the complete image of the element, okay, let's move ahead, next, see the next concept is independent and dependent variable, this is very easy, see, independent means you know the one which does not need anyone and dependent means the one on which I am talking, so an independent variable is a variable that represents a quantity which is being a set and manipulated in an experiment, okay, I am explaining it now, see an example, secondly, what is a dependent variable, it represents a quantity whose value depends on the independent variable, okay, now one of its For example, look, we have written area here as Paa s, area of ​​circle, we have written Paa s, now the a here, what is it, is it dependent, what is it, a depends on the value of r, now this a on which it depends will become independent, okay, and the a which is dependent will become dependent, tell me, did you understand, see, I will explain it again, area equals Paa s, you have given a value, the area here, if I change the radius from one to two or three, then what will happen to the area here, it will change, what will happen to you, the area will also change, do you understand what I am trying to say, so what I mean to say is that if this area is depending on what, it is depending on the radius, it means what has become of the area i.e. a, it has become a dependent variable and there is no difference on the radius, it is coming on its own, it is increasing as per its wish, it is decreasing as per its wish, so what will happen to r, it will become an independent variable, tell me, did you understand So here also it is saying that the dependent variable represents the quantity whose value depends on the independent variable. The value of a is dependent on r and r is independent. Do you understand this? And what is an independent variable? It is a variable that represents a quantity which is being set and manipulated in experiments. We can manipulate it accordingly by replacing a with a, so that it becomes independent. It is clear that you have the independent variable and the dependent variable. Move ahead. Okay, next look in the context of a function, the independent variable is the input. It is the same thing. What is the independent variable? It is the input in which we are putting the value inside the input and what is the dependent variable? It is the output which we are getting. It is simple, isn't it? It is an input to the function and the dependent variable is the output of the function. It is clear to everyone here. For example, if y is a function of fxxx.pro x, the value of y depends on x. Next, look, now comes the domain and range. This is also a great concept. Look, let's call it a function for x, okay, the set of x, here for x, 3, 4, 5, whatever happens, what do we call them, it is called domain. The values ​​written in y are 7, 8, 9, 10, right? All these values, what do you have, you have understood the domain till here. Now, suppose I am matching these here, one becomes seven, two becomes eight, i becomes 10, 11 also takes 12, four becomes 10, here 11 is left, 12 is left. Now, whatever values ​​I have, there is an arrow coming here, right? I mean, whatever values ​​are there in my answer, what do we call them, they are called range. Did you understand what I am trying to say, there are some values ​​which are left in this, some in the answer. The values ​​that are left in the answer will not be included in the range, so the set consists of all the images, I told you, the function that you are getting y, right, what is your image, this is your image, what is your image that is coming out in the answer, the image of the element x under the function f is called range of f, okay, what is this called, it is called range of f, or your image, so the range of function will be the function for all x belongs to x, all the x that belong to the set, and that value will be the range of a, tell me, did you understand, is it clear, note one condition, now see, note here, this range is in a way a subset of the domain, this range of yours is a subset of whose domain, but it can also happen that in some case, all the values ​​that are coming out of the range are equal to the codomain, yes, it can happen, so should I add one more small symbol below here, yes, I can do it, that is That the range is both a subset of the domain and is also equal, did you understand what I am trying to say, I mean your range can be a subset of the codomain and can also be equal, both the conditions can be there, okay it cannot be greater than that, but the range of a is a subset of y codomain and may not be equal to va, okay it may be, it may not be equal, so I will use this one for that, tell me the subset one, did you understand, is it clear, all the students do not have any doubt, right? No, sir, it is not, okay, so this range belongs to a subset, it has become equal, it is clear for the codomain, here it is clear in the concept of range and domain, okay, let's see further, now I will give you an example of this, now you will understand it better, now there is an example, there are two sets for x and v, if you have seen all these examples, then I am telling you that you will find ec9 on your own, you do not need anyone, x and v are two sets, both sets See the value comfortably, I am giving you a function of my own free will, you don't have to make it like this, I am giving it to you, now you will say sir I can create a lot of relation in it, yes many relation will be created, see first we will multiply one by zero, if I create a relation then I use x multiplied by y, so I will first write one with zero, then I will write one with one, then I will write one with two, one with three, do this then I will write two, then three, do this, there will be a lot of it, I am giving you a function myself, where I am writing 1 and 20 37 49 5 13, okay, there is a function x2y because now tell me is this a function, I have given it to you, now I am asking you, tell me is this a function, first of all I will make its diagram, first of all do this mapping, it is easiest to understand with this mapping, see here we will write those elements which are inside x All the elements are given in the set 1 2 3 4 5 here we will write those which are given in the y set okay all the elements you must know what this is called domain what this is called codomain now see here in the function one to one is added two one to one is added then two to zero is added two three to one is added two four to nine is added two five to 13 is added two now you see whether two images are coming for any element are coming two images are coming for any element are they not unique for all what is a unique image for all what is a unique image for all if this is a unique image then we can call this a function see x to y because each element of x has a unique image in va simple if I ask you for the range in this then what do you have to tell you on whichever arrows are coming in va they will be only ranges like zero will be one will be sev, it will not be 13 tell me did you understand note the sum element of va are not associated with any element of x, there is no f due to this, there are many such elements left inside the va which are not related with x, so there is no problem, what are you going to do, leave them, they will remain inside the codomain as it is, okay, understood, you have to understand it lightly, there is no problem, right, let's move ahead, our second example, one more, this time I have created another function, the set is the same for you, I have created another one from my side, one and three and three and four and four from four, five, okay, now I will draw the same arrow for this also, from one I sent three, then from two also I sent three, okay, no problem, then from three I sent five, okay, then from four I sent sine, then from five again f, you will note one thing, here this time an image is coming for two elements, it is coming, it is not coming, see here an image is coming for both these elements, f and sine is coming for this one, okay, this is also your function, why have I told you that every each The element of x should not have two images what am I saying listen again it should not have two images but any two can have one image do you understand what I am trying to say any two can have one image no problem brother if we look at one then there are three if we look at two then also there are three so it is unique brother there are not two images for one there is only one image its brother for one the answer is three and for two also three is coming the answer so still the concept of unique image is the same do you understand what I am trying to say this time your range will be 3f and sen something is sitting in my mind what I am trying to say there is a difference between the two things in the concept of unique image two answers cannot come for one but there can be one answer for two it is clear note that in the second components the second component is repeating here it is okay here it is repeating in this also first It will not do the component, right? First component will not do it. Now if you assume this happens, one and three and one and four, now the matter gets messed up, this is the past, the matter is only answer of one and answer of one is only four, this is not a unique image here. Yes, there are two images, tell me, did you understand? You guys should understand the concept of unique image a little more carefully. I am again asking you to focus a little on the concept. So, I am telling you that the easiest chapter function is going to be, I am telling you that within three days, you can complete the chapter, each exercise each day, you just have to focus and you have to pick up all the lectures from the very beginning. Okay, so let's move ahead. Next, let's see, I have more examples of this. Okay, I will make you read the entire ML Aggarwal book like this. I mean, I will not leave any doubt for you in any paper from anywhere in that book. Okay, let's move ahead. The second example is done. Look, this is also a good example. This time I am making a set. One and two and five and four and five and zero. Okay, first, join them. I added five to one. I added seven to two. I added nine to four. I added zero to five. Sir, it is a unique image. Yes, it is a unique image. Is there any problem? No, but one element is left in x. This function will not work because the element 3x has no image. Every element must have an image. Every element must have an image. A unique image. If any elements are left here, then there is no problem, but they should not be left here. Did you understand what I am trying to say? Okay, so everything else was fine, but this three messed things up. So this is also not a function. You will get such questions for checking. They will give you diagrams in the exam and will ask you to tell which of these is a function and which is not. Do you understand what I am trying to say? So, let's move ahead and take the next one. Let's come to the fourth one here. Let x be x. 1 2 3 4 5. Okay, and 0 1 2 3 5 7. Okay, I have created the same function again. 1 1 2 2 3 3 5 4 7 5 11. Okay, see here, everyone has it, but this one is getting two images. If you do my calculations like this. If you take the first line, I had said that the first component should not be the same. As soon as the first component is the same, the matter is wrong. As soon as the first component is the same, the matter is wrong. So, for one, one image is coming here. For this one, two, that is, one image is not there, then this will also not be a function. Because the different pairs, one and two, have the same first component. You can understand it like this and you can also understand it from the image. Element one of x has two different images. Two different images are coming. Tell me, did you understand it? Is it clear? You have to understand it absolutely lightly, comfortably. There is no hurry. The more easy- going you are, the more you will strengthen your concepts in the chapter. Okay, you just have to be a little patient and the lectures have to go on. If you sit and take the lectures for two-three hours, then there is no problem for you. Let's move ahead. Next, let's summarize all this and make some points. I will explain to you in these points. I think I have given you the entire functional concept. You must have taken it well, there are a few points, I am explaining them by summarizing them, you guys please see them, where did the second go, fourth, that's it, okay, main features of function, now all the points of this function that I have mentioned, summarizing them, you have understood all the points, see, the same things have come, every x belongs to x, there exists a unique element x, y belongs to y, it is true that y = f, this I explained to you in the beginning, no element of x can have more than one image to y, this also I explained to you, there may be an element of y which is not associated with any element of x, there may be an element of y which has no association, which has no element with one, okay, it may not be possible, there cannot be a distinct element of one, it has the same image, it means that two elements, like two and three both are getting the same image, it is possible, all these points are summarizing completely. You have read the function, are you understanding it, function A is determined one, f is none for all, isn't it, function A will be formed only when f is none for all x, all the x's belong to it, there should not be anything with x which does not belong, tell me did you understand, so this is the summary, whatever you have read, see, if you want to read the function, then it is better if you guys are watching this video, if you whoever is in your friends group, if you end the chapter directly, then you suggest them that please watch this lecture once, if they have to take this introduction part, then I am telling you that you guys will make it so easy, you will make it so easy, it is not to this extent, I have taken every chapter from the beginning, I have taken the introduction part before every chapter, every exercise, I have given examples and let me tell you the most amazing thing about this channel is that the Applied Mathematics that you are getting here is ML Agarwal's new book, it is absolutely a new edition. According to that, the first thing, second thing is here, if you look, if you pick up any chapter, go to the playlist, go to the home section and pick up the playlist of any chapter, then chapter number one is there, the thumbnails given with that chapter number one have page numbers written in them, I see how wonderful it is, this took a little effort of mine but the work has become very easy for you people, you can search the page number, the page numbers are written in it, you have a doubt sir, I want to see this example, my example number one, now there are many example number one in the book, you search for its page number, then go to the playlist of my chapter number one and look at that page number, you will see the page number, easily you open the example with that page number, you open this exercise, the page number will directly match with your book, so you don't have to worry about anything and like sir how are you doing, where is which case, I am not able to find it, then this is the situation I wo n't let you come, just accept it. What you have to do is follow a little bit. You have to take the lectures from the beginning properly. You will read the chapters through and through. Only then can you save a little bit from the coaching and all that you need. And this can be done for you on my channel. The rest is not that hard. I am making everything available. If possible, I will try to arrange live sessions etc. But for now, my main motive for you guys is that before November, I should mind your syllabus and provide you with lectures here so that you guys don't face much problem in the middle of the exam. Okay, till here it is clear here. So this is complete chapter number nine - function introduction part for exercise 99.1. And if you have any doubts, do tell me in the comment section. And please, friend, don't be stingy in liking, sharing and subscribing. Please, please, if it is taking a little more effort, then the support etc. that I get through you guys will be very good. I want to call the teachers here, in my view there are some teachers who can make other subjects also excellent for you people, but for that they need a little help from you people and that help can be provided only when you do not show any hesitation in liking, sharing and subscribing, press the bell icon so that the notifications of the next lecture or any other information will reach you first, so we will meet next with example number one, till then goodbye and if you people are having exams then best of luck ok