Function symmetry introduction

- [Instructor] You've likely heard the concept of even and odd numbers, and what we're going to do in this video is think about even and odd functions. And as you can see or as you will see, there's a little bit of a parallel between the two, but there's also some differences. So let's first think about what an even function is. One way to think about an even function is that if you were to flip it over the y-axis, that the function looks the same. So here's a classic example of an even function. It would be this right over here, your classic parabola where your vertex is on the y-axis. This is an even function. So this one is maybe the graph of f of x is equal to x squared. And notice, if you were to flip it over the y-axis, you're going to get the exact same graph. Now, a way that we can talk about that mathematically, and we've talked about this when we introduced the idea of reflection, to say that a function is equal to its reflection over the y-axis, that's just saying that f of x is equal to f of negative x. Because if you were to replace your x's with a negative x, that flips your function over the y-axis. Now, what about odd functions? So odd functions, you get the same function if you flip over the y- and the x-axes. So let me draw a classic example of an odd function. Our classic example would be f of x is equal to x to the third, is equal to x to the third, and it looks something like this. So notice, if you were to flip first over the y-axis, you would get something that looks like this. So I'll do it as a dotted line. If you were to flip just over the y-axis, it would look like this. And then if you were to flip that over the x-axis, well, then you're going to get the same function again. Now, how would we write this down mathematically? Well, that means that our function is equivalent to not only flipping it over the y-axis, which would be f of negative x, but then flipping that over the x-axis, which is just taking the negative of that. So this is doing two flips. So some of you might be noticing a pattern or think you might be on the verge of seeing a pattern that connects the words even and odd with the notions that we know from earlier in our mathematical lives. I've just shown you an even function where the exponent is an even number, and I've just showed you an odd function where the exponent is an odd number. Now, I encourage you to try out many, many more polynomials and try out the exponents, but it turns out that if you just have f of x is equal to, if you just have f of x is equal to x to the n, then this is going to be an even function if n is even, and it's going to an odd function if n is odd. So that's one connection. Now, some of you are thinking, "Wait, but there seem to be a lot of functions "that are neither even nor odd." And that is indeed the case. For example, if you just had the graph x squared plus two, this right over here is still going to be even. 'Cause if you flip it over, you have the symmetry around the y-axis. You're going to get back to itself. But if you had x minus two squared, which looks like this, x minus two, that would shift two to the right, it'll look like that. That is no longer even. Because notice, if you flip it over the y-axis, you're no longer getting the same function. So it's not just the exponent. It also matters on the structure of the expression itself. If you have something very simple, like just x to the n, well, then that could be or that would be even or odd depending on what your n is. Similarly, if we were to shift this f of x, if we were to even shift it up, it's no longer, it is no longer, so if this is x to the third, let's say, plus three, this is no longer odd. Because you flip it over once, you get right over there. But then you flip it again, you're going to get this. You're going to get something like this. So you're no longer back to your original function. Now, an interesting thing to think about, can you imagine a function that is both even and odd? So I encourage you to pause that video, or pause the video and try to think about it. Is there a function where f of x is equal to f of negative x and f of x is equal to the negative of f of negative x? Well, I'll give you a hint, or actually I'll just give you the answer. Imagine if f of x is just equal to the constant zero. Notice, this thing is just a horizontal line, just like that, at y is equal to zero. And if you flip it over the y-axis, you get back to where it was before. Then if you flip it over the x-axis, again, then you're still back to where you were before. So this over here is both even and odd, a very interesting case.