Introduction to Odd and Even Functions

INSTRUCTOR: Welcome to an introductory lesson on even and odd functions. There are two main ways to determine if a function is even or odd. One is an algebraic method, and the other is a graphical method. 

A function is even if algebraically f of x is equal to f of negative x and graphically the graph has symmetry across the y-axis. And if we're working with a polynomial function, a polynomial function will have all even exponents on the variables for odd functions. Algebraically, negative f of x is equal to f of negative x, and graphically the graph has rotational symmetry about the origin, which means the graph remains the same after a rotation of 180 degrees. And a polynomial function that is odd will have all odd exponents on the variable. Let's take a look at two basic examples to get an idea of what's happening here. 

We'll consider f of x equals x squared and f of x equals x cubed. Looking at the graph of f of x equals x squared, notice that if we were to fold this graph across the y-axis, it would match up perfectly with the other half, which means this graph has symmetry across the y-axis, and therefore f of x equals x squared is an even function. If we want to show that it's even algebraically, we would have to show that f of x is equal to f of negative x. Let's go ahead and do that. 

Again, to show this function is even algebraically, we want to show that f of x is equal to f of negative x. Well, we're given f of x, so let's determine f of negative x. To determine f of negative x, the input into our function is going to be negative x instead of just x, so we would have negative x squared. Well, a negative times a negative is positive, so this is equal to x squared. So f of negative x is equal to x squared, and we already know f of x is equal to x squared. So we've just shown algebraically the function is even. So since f of x equals f of negative x, the function is even. 

One more thing we could mention about this basic function is it's a polynomial function, and all the exponents or even. Therefore, it's also an even function. Now let's consider f of x equals x cubed. To determine graphically if this is an odd function, the graph would have to have rotational symmetry about the origin, which means if you rotate this 180 degrees, or half a turn about the origin, the graph should remain unchanged. 

So, if we rotate this 180 degrees, notice how this piece here in the first quadrant would match up perfectly with this piece here in the third quadrant. And the piece in the third quadrant would end up in the first quadrant, and the graph would look exactly the same. Therefore, f of x equals x cubed is an odd function graphically, but to verify this algebraically we need to show that negative f of x is equal to f of negative x. And let's go ahead and do that as well. 

So we want to show negative f of x is going to be equal to f negative x. Let's start with the right side and determine f of negative x. So we're going to replace x with negative x in our function, so we'd have negative x to the third. Well, here we'd have three negatives being multiplied together, so this ends up being negative x cubed. 

And then for negative f of x, this means we're going to change the sign of our function. So if we're taking the opposite of f of x, we would take the opposite of the right side as well. So this would equal negative x cubed, and notice that negative f of x is equal to f of negative x. They're both negative x cubed, therefore we verify that this function is odd. 

We're going to stop here on this introductory video. We'll take a look at several more examples of determining if a given function is even or odd in the next few videos. So if you want to, you can follow the links to the additional examples on the screen. I hope you found this helpful.