Even and Odd Functions

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle first. With numbers, odd times odd is always odd, isn't it? Three times five, fifteen, odd. Now take two odd functions, x and sine x, and multiply them. So x sine x should be odd, right? Hmm. It is actually even! But how? Today we find out what even and odd really mean for functions. And look, in JEE, this symmetry idea comes up again and again. So, let's begin.

0:32Even and odd: the definition

So, first one condition. The domain D must be symmetric: if x is in the domain, then minus x is also in it. On screen, the curly E means, belongs to, and the double arrow means, implies. Why this condition? Because we compare f of x with f of minus x, so both must exist. Clear? Now, replace x by minus x. What can happen to the output? Simplest case: nothing changes at all! f of minus x equals f of x, for every x in the domain. Such a function is called even. Look at the left picture. The two coloured dots, at x and at minus x, sit at the same height. So the left half of the graph is the mirror image of the right half. Even means symmetric about the y-axis. With me? Second case: the output flips its sign. f of minus x equals minus f of x, for every x in the domain. Such a function is called odd. Now the right picture. At x the dot is up here, and at minus x it is the same distance down there. The dotted line joining them passes through the origin, right in the middle. So odd means symmetric about the origin, okay?

1:47Products: even like plus, odd like minus

Alright, now let's combine functions. Sums are easy. Even plus or minus even stays even, and odd plus or minus odd stays odd. The little plus-minus sign means, read it once with plus, once with minus. Now our puzzle. If f and g are both odd, is the product f g even, or odd? Odd, you said? Like numbers? Let's check, shall we? Put minus x in the product. f of minus x is minus f of x, and g of minus x is minus g of x. Minus times minus is plus! So the product comes back unchanged. Even! So odd times odd is even, just like even times even. That is our hook: x is odd, sine x is odd, so x sine x is even. And even times odd? One minus sign survives, so it is odd. Topper's trick for products: think of even as plus, and odd as minus. Plus times plus, plus. Minus times minus, plus. Plus times minus, minus. That's it! But careful, this trick is for products only, not for sums, okay?

2:50Composition: follow the minus sign

Now composition, f of g of x. The small circle on screen means composition. If the inside function g is even, f of g is even, whatever f is. Because g of minus x equals g of x, so f gets the same input, right? If f and g are both odd, g of minus x is minus g of x, and then the odd f pulls that minus sign outside. So f of g is odd. And if f is even and g is odd, the even f simply ignores that minus sign. So f of g is even. Got it? Just follow the minus sign from inside to outside.

3:24Two special facts

Now, a quick question. Can one function be even and odd at the same time? Yes, it can, but only one function! Let's see why. Even means f of minus x equals f of x, odd means it equals minus f of x. Together, f of x equals minus f of x. The only number equal to its own negative is zero. So the only such function is the zero function, f of x equals zero, okay? Second fact. If f is odd and defined at zero, put x equals zero. Minus zero is zero, so f of zero equals minus f of zero. So f of zero must be zero. Look at the right picture: the odd curve passes through the origin. Fast exam check: if f of zero is not zero, the function cannot be odd!

4:14Every function = even part + odd part

Now the most beautiful result here. Every function on a symmetric domain splits into an even part plus an odd part. The even part is f of x plus f of minus x, over two. Replace x by minus x: what changes? Nothing, exactly! The two terms just swap places. The odd part is f of x minus f of minus x, over two. Replace x by minus x, the terms swap, and the sign flips. Odd! Add the two parts: the f of minus x terms cancel, and you get back f of x. It works out perfectly. And this split is unique. For a polynomial, simply put the even powers in the even part, and the odd powers in the odd part, okay?

5:00Example 1: even, odd or neither for six functions

Right, Example one. Classify these six: x to the four plus x squared; x cubed minus x; x squared plus x; mod x; x sine x; and e to the x. Pause the video, and try it first. Done? Okay, let's check. x to the four plus x squared has only even powers, so putting minus x changes nothing. Even. x cubed minus x has only odd powers, so every term flips. Odd. Now x squared plus x. x squared is even, x is odd. So the sum is even? Or odd? Hmm, neither rule fits, right? This is the JEE trap: students think every function must be even or odd. No! Test one point. f of one is two, but f of minus one is zero, neither two nor minus two. So: neither. And that is perfectly allowed! Mod of minus x equals mod x, so mod x is even. And x sine x, our hook: odd times odd, even! Last, e to the x. The slash through the equals sign means, not equal: e to the minus x is neither e to the x, nor minus e to the x. So, neither. Our quick check agrees: e to the zero is one, not zero, so it can never be odd.

6:20Example 2: JEE Main 2019, even part of a to the power x

Now a JEE Main twenty nineteen question. f of x equals a to the power x, with a positive, written as f one plus f two, where f one is even and f two is odd. Express f one of x plus y, plus f one of x minus y, in terms of f one. Pause and try. Hint: find f one first. Done? Okay, let's check. f one is the even part: f of x plus f of minus x, over two. So f one of x is a to the x, plus a to the minus x, over two. Put x plus y, then x minus y, and add. On top, four terms: a to the x plus y, a to the minus x minus y, a to the x minus y, and a to the minus x plus y, all over two. Can you factorise these four? Yes, we can! They are a to the x plus a to the minus x, times a to the y plus a to the minus y, still over two. Each bracket looks like f one, so the answer is f one of x times f one of y, right? No, no, careful! f one has a two at the bottom, so each bracket is two times f one. Two times two, over two, leaves two. Final answer: two f one of x, f one of y. Clear?

7:43Quick revision: f(-x) test, symmetry and product rules

So, quick revision. Even: f of minus x equals f of x, a mirror image in the y-axis. Odd: f of minus x equals minus f of x, symmetric about the origin. And for products, even is like plus, odd is like minus. Now go and try the practice problems on this page. All the best!