 Okay, so we are going to talk about even odd functions over here. Even odd functions. What's an even function? What is an even function? An even function and even function is a function. This satisfies this functional equation. Yes, this is actually a functional equation. Okay, so any function which satisfies this functional equation is said to be an even function. Okay, examples of this would be many function that you have come across like x square. Okay, cos x function. Okay, mod x function. Okay, these are examples of even function. As you can see, if you replace your x with a minus x, nothing happens to the function. So this is an example of an even function. Okay, similarly, what's an odd function? An odd function is the one which satisfies this functional equation. Okay, typical example of this would be your x cube function, sine x function. Okay, or you can say any kind of a polynomial or any kind of an algebraic equation having only odd powers of x. Okay, tan x function. These are all examples of odd functions. From the function definition, it is clear that even function, the value of y doesn't change if and if you change x to minus x. That means the function graph is exactly symmetrical about the y axis. It's symmetrical about y axis. But here if you see, the value of y flips if your x sign flips. In other words, if you change the sign of x, even the sign of y gets changed. In such cases, we say the graph is symmetrical about origin. Okay, so whatever you draw in the first quadrant, you will draw the same thing in the third quadrant. And whatever you draw in the second quadrant, the same thing you will draw in the fourth quadrant. That is what is meant about symmetrical about origin. Symmetrical about y axis, we all know it's mirror image about the y axis. Okay, now let us look into some properties of even odd function and then we'll start solving questions on the same properties of even odd functions. The first property that I would like you to understand is a function can be neither. Why I chose this property as my first property? Because many people have this wrong notion that every function in this world will either fall under even function or fall under odd function. Just like every integer is either even or odd, it doesn't work the same for functions. Functions can be neither also. So a typical example I would give you is something like x square plus x cube. If you change your x with a minus x, it neither generates f of x, it neither generates f of x, it neither generates negative f of x. So it is neither even or odd. So it is not compulsory that every function in this world has to be even or odd. It can be neither also. Okay, second property, any function, any function, any real valued function can be expressed as, can be expressed as sum of an even and an odd function. Stock resemblance with matrices, any matrix, any matrix square matrix can be expressed as a sum of a symmetric and skew symmetric matrix in the same way. Now why does this work? It works because let's say I have a function f of x. Okay, now this could be even, this could be odd. Correct. Doesn't matter. It could be neither also. Doesn't matter. Can I write this function like this? f of x plus f of minus x by 2 and f of x minus f of minus x by 2. Okay, of course you would agree with this because if you open the brackets here, f of x by 2 and f of x by 2 will add up to give you f of x, rest of the term gets cancelled. Now what I'm claiming here is that this function, let me name it as h of x is actually an even function. And this function, let me call this as g of x is actually an odd function. That's what I'm trying to claim. So let's check whether they're actually even or not. So if you want to check this is even or odd, you just change your x with a minus x. When you do that, what will happen? This will become f of minus x f of x by 2. Doesn't it give you the same function back? Absolutely. Okay. Yes, hello. Advik has a question sir. Is this what the expansion of e to the power x works on? I didn't get that. e to the power x expansion comes from Maclaurin series. Okay, so that means h of x is an even function. Is an even function. Okay, is an even function. Similarly, if you check with g of x, in g of x, if you replace your x with a minus x, it will become f of minus x minus f of x by 2, which is clearly negative of f of x minus f of minus x by 2, which is clearly negative of g of x, which implies that g of x is an odd function. Okay. At the same time, I would like you to know one more property so that we can justify when your function f is exactly even or exactly odd. Remember, zero function is both even and odd. This is both even and odd. Okay, so zero is such a function which is both even and odd. That is why if you have, let's say just an x squared, if you just take an x squared, x squared can be written as x squared plus zero, and then x squared becomes even and zero becomes odd. If you write x cube, x cube can be written as zero plus x cube, then zero becomes odd, sorry, even and x cube becomes an odd. Okay, so zero can play both the rules. So if your f of x is purely even or purely odd, then zero can be used to justify that this property will hold even good at that time. Is that correct? Is that fine? Okay, so property number three, I hope you have written it down. I would like to go to the next phase. So if you want to copy anything, please do so immediately. Yes, sir. Okay. Next property, which is probably number four, every even function is a many one function. Every even function, even function is a many one function. Of course, for all x belonging to the domain of the function. Okay. If you take any even, even function, let's say cos x function. Okay. In fact, let's take x square. Those all known to be many one function. That means if you draw a horizontal line that will definitely cut the function at two or more than two points. Okay. But can we say the fact that all odd functions are one one functions that we cannot say a clear cut example for that would be sine x, which violates that rule. Sinex is a many one function and still it is a odd function. Still it is an odd function. So if this is true, do not think that every odd function is a one one function. No. Okay. Sinex, tan x, they are very classic examples. Cosic x, they're classic examples of odd functions, which are not one one, which are not one one. Okay. Next, first derivative. Let me write it like this. Yeah, let me write first derivative only first derivative of an even function. Even function will always give you an odd function and vice versa. First derivative of odd function will always give you an even function. Why is this true? Can somebody justify this? Can somebody prove this? Why is this true? An example. No, no, no, no. Proving is not an example. Prove it, prove it. That means any even function if you take, if you differentiate it, it will always get an odd function. So we can take the definition and just differentiate that. Right. So let us say if I take an even function, that means a function which satisfies this criteria. Okay. So let's say f is an even function. Let's say f is even function, which implies it satisfies this criteria. If you differentiate it, and let's say I call this as some g of x. Okay. What do you get here? What do you get here? You get minus of f dash minus x. Isn't it? Minus of f dash x. Correct. Yes or no? And what is this actually? This is minus of g dash minus x. Sorry, minus of g of minus x, isn't it? Because if you're calling this as your g of x, this term, put a minus here and then minus will come over here, replace x with a minus x. This will become here. So indirectly, what are you trying to claim? You're trying to claim that g of x is negative of g of minus x. That means g of minus x is negative of g of x. Which means g of x will be an odd function. Got the point? Yes or no? Graphically, how would you prove it? So I was thinking like an even function will have the same value, but on the other side, the slope will be negative. It'll be the same thing, but negative. And since the slope is the derivative of this, that means since it's changing sign on both sides, that means it has to be odd. Okay, so basically what he's trying to say is that let's say there's an even function like this. If you take any point here, let's say x and sketch a tangent here and take any point minus x and sketch a tangent over here. Let's say if this has a slope of k, then this will have a slope of minus k. That means the derivative will have, sorry, minus k. Yes, absolutely correct. That's a way to guess it. Okay. Similarly, you can prove the other way around also that the derivative of an odd function will always be even. Okay. Let's take the sixth property. Let's take the sixth property. Six property I would like to make a box here. G sum difference product. Okay. So this is your f of x. This is your g of x column. This is your sum of FNG. This is difference of FNG. This is product of FNG. This is quotient of FNG and this is composition of FNG. Okay. Let's write what will be the nature of the sum difference product or quotient when both are even. So it is even. This one even. Any doubt regarding this sum of two even functions will be even. What about the difference? Even. Even. What about the product? Even. Quotient. Quotient. Even. Any doubt with quotient being even. Let me know if you have, if you're doubtful about any one of them. Okay. This is composition. That will also be even. Okay. Now some people may want to know about the composition. See, let's say I take h of x to be f of x into f of x g of x. f of g of x. My bad. If you replace your x with a minus x, it will become f of g of minus x. But since g is even, it will immediately convert g of minus x as g of x giving you h of x back. Okay. So h of negative x and h of x gives you the same expression. That means it is also even. So can you tell for the quotient also? Yeah. Quotient also, let's say h of x is f of x by g of x. If you replace your x with a minus x, it will become f of minus x by g of minus x. And since both of them are even, it will come back to the original shape. That means it will come back to h of x. Okay. What about even and odd? You can't comment about it. You can't comment. So say neither. Okay. It will be neither even nor odd. Same with difference. Let me write it in white. What about the product? Odd. Very good. What about the quotient? Odd. Odd. What about the composition? The even. Even. Absolutely. It will be even. Anybody who wants to know why it is even, I will quickly discuss this here. See, I have already written it over here. Let's say your f is even and g is odd. Then what will happen? At this step, there will be a minus sign created over here. But remember f is an even function. So it will not take, it will not care about this negative in its input. It will just treat it as f of g of x. That means you are back to the same function. And if you are back to the same function, it means it is even. Odd even. Neither even. Neither. Neither. Product. Odd. Odd. This. Odd. This. Even. Even. Absolutely correct. What about odd odd? Some difference. Odd odd. Product. Product even. Even. Quotient. Even. Composition. Composition will be odd for the first time. Okay. So when both the functions are odd, composition will also be odd. Let's check. Let's check. So if h of x is f of g of x, let's change x with a minus x. It will become f of g of minus x. Now remember g is an odd function. So immediately minus sign will come out. And even f is an odd function. So immediately minus sign will come out here also giving you negative of h of x. Yes. It is an odd function. Please note down this table. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. No. odd okay and continuous also I should add and continuous then f0 will definitely be 0 are you getting my point in other words if f of x is an odd continuous function is an odd such that f of 0 is not 0 then f of x cannot be odd sorry if it is a continuous function and f of x f of 0 is not equal to 0 then f of x cannot be odd sorry by mistake I wrote an odd in the in the beginning if it's an even function it can be 0 or anything else sorry yeah function f0 can be anything else right for example x square plus 2 at 0 it can be 2 also correct but if you have an odd function which is continuous and it is passing through 0 that means its value of y should be 0 also at that position that means it should pass through origin okay so a continuous function which is odd and 0 is a part of its domain then it will always pass through the origin so why does it have to be continuous so what it would be automatically continuous if it's odd and exist at 0 oh no no no no let's say if I have a if I have a situation like this and let's say this value is here yeah so yeah so if there is a let's say there is a value missing here then what will happen it is odd but it is not continuous at 0 okay so this will create a problem in the same way there is a similar funda for even also for even function we say that let me write this as a let me write this as be a differentiable even function if f of x is a differentiable even function differentiable even function okay differentiable even function means differentiable everywhere or you can say it is differentiable in the domain which includes 0 then f dash 0 would be 0 okay x is a part of its domain differentiable I should write yeah if f of x is a differentiable even function where 0 is where 0 is included in its domain then f dash 0 will be 0 get in the point alternately you can say alternately you can say if for a differentiable function if for a differentiable function having 0 in its domain function with 0 belonging to the domain if f dash x is not equal to 0 then f of x cannot be even cannot be even it is just a negation of the same thing now in the interest of time I'll give you some function in fact I we have 8 minutes left will give you some function and I'll ask you to categorize it under even odd or neither let's take a question let's take a simple question log of 1 plus x by 1 minus x second let's say log of x plus under root of 1 plus x third one let's say e to the power x e to the power minus x by 2 fourth one x sine x plus tan x upon gif of x plus pi by pi minus half so this represents gif okay where where x is not a multiple of pi x is not a multiple of pi let's start I think these are all easy ones so let me start with the first one first one please put your answer as one and then write e or o or even odd whatever you want to write first one is o that's absolutely correct because if you do f of minus x it becomes log of 1 minus x by 1 plus x which is actually negative of log 1 plus x by 1 minus x so yes absolutely correct this function is an odd function second one second one neither neither neither okay okay let's check let's check in second one if you take f of x as if you change your x as minus x you'll get log of negative x plus under root 1 plus x square okay now just try to rationalize it just try to rationalize it so let me write it like this log under root 1 plus x square minus x okay let's try to rationalize the inside part something like this so the numerator will become if i'm not mistaken a square minus b square that will lead to a 1 denominator you can write it as x under root of 1 plus x square which is actually negative of log x plus under root 1 plus x square which is negative of f of x clearly indicating that even the second one is odd so people who wrote neither it's wrong it's actually an odd function okay it's actually sine hyperbolic inverse x anyways this is pretty simple if you replace x with a minus x nothing happens to the function so this is an even function no problem with that what about the fourth one i would like you to work on the fourth one x is not a multiple of pi be aware of that okay ruchir okay two people have responded so far with respect to the fourth one okay rasdeep very good chelo guys we'll discuss this in the interest of time because we don't have much time left let's try to let's try to first write it in a simpler way this will be x sine x plus tan x okay this will be x by pi plus 1 minus half right now we all know that when you have a gif term with an integer in between integer comes out of the gif symbol okay so i hope everybody is well aware of this property remember we had done this in our theory of equations right so x plus n or minus n where n is an integer okay can be written as gif of x plus minus n correct okay now this is your function think like this now on this we have to check what happens if i replace x with a minus x if i replace x with a minus x numerator will become minus x sine of minus x tan of minus x by gif of minus x by pi plus half numerator nothing will happen because a minus sine minus x will be minus of sine x and tan minus x will be minus of tan x because sine tan both are known to be odd functions so odd odd is even so nothing will happen to the numerator function let's check the denominator function now i don't know how many of you remember this gif property that if x is not an integer gif of minus x is minus 1 minus gif of x how many of you remember this property some of you do okay good so this can be written as minus 1 minus gif of x by pi plus a half okay now people are realizing that they did a mistake so this will be this will be negative gif of x by pi negative half so you can take a negative up if you want which is which is negative of x sine x plus tan x upon gif of x plus pi sorry gif of x by pi plus half okay i'll not put a bracket here else it'll appear as if i put in a gif over here so which is actually actually if you see this function's negative isn't it this functions which i'm showing with my cursor is negative of that so this will be negative of f of x okay clearly indicating that it's an odd function clearly indicating that it's an odd function okay well we have more agendas on even odd functions next class when you meet we'll do some bit of more problems probably some trickier ones and then we'll start with periodic functions functional equations to finish off this functions chapter and then we'll start with definite okay thank you bye bye take care stay safe thank you thank you sir Dhruv take care of yourself bye who's making that okay bye