 When it comes to studying functions, it'll be useful to study what we call the symmetry of a function. And we're used to notions of symmetry, maybe from past geometry classes and such. And so what I want to quickly do is introduce two notions of symmetry that we will often discuss as we talk about functions this semester. The first type of symmetry is what we call even symmetry. And a function we say is even if it's symmetric with respect to the y-axis. And so as you can see to the graph illustrated to the right right here, if you look at the y-axis, this is a line of symmetry. That is, if you were to reflect the graph across the y-axis, you'll get the exact same picture, right? It's just a mere image of itself when you look across the y-axis. This is what we refer to as an even function. An odd function is a different type of symmetry here. We say a function is odd if it's symmetric with respect to the origin. This one's a little bit more funky to understand here, but symmetry with respect to the origin, you can think of it in one of two ways. One way is if you take any point and you take the line that goes through the origin, follow that same distance to the other side, and you'll find a point on the graph. So you have this like axial line right here that acts as symmetry here. If you don't like that, you can think of the following idea. You take your graph and you rotate it a half spin that is you rotated 180 degrees around the origin. And you're going to see that when you spin this thing around the origin, a half spin, you get the exact same picture again. This is what we call an odd symmetric function or just an odd function. The names even and odd might be a little bit weird because like evens and odd aren't those numbers like two is even and three is odd. It turns out that these names even odd have something to do with numbers and that parity that we talked about a moment ago. But we'll make that a little bit more explicit later on when we talk about symmetry with respect to algebraic functions. For the time being, I just want you to know that a function is even if it's symmetric with respect to the y-axis, it's odd if it's symmetric with respect to the origin. And I want you to be able to visually identify these two type of symmetries if you look at the graph of function. Now, I should also mention that it's not actually possible for a non-zero function to be symmetric with respect to the x-axis. Because you might wonder like, well, we have the y-axis right here. Why can't we read the symmetric to the x-axis? If your function was symmetric to the x-axis, that means when you reflect across the x-axis, you should find another point, right? But that would violate the vertical line test. So a non-zero function cannot pass the vertical, sorry, a non-zero function cannot be symmetric with respect to the x-axis. So we're not going to worry about that type of symmetry when it comes to functions. I should also mention that it's not possible for a function to be both even and odd simultaneously. Because if it was even and odd, you could actually argue it would have to be symmetric with respect to the x-axis, which then gets us in the same problem we were a moment ago. And I should also mention that it's possible for a function to be neither even nor odd. It's not a requirement when it comes to functions. Like if we had a graph that did something like the following, do-do-do-do-do-do, right? This function is neither symmetric with respect to the origin or to the y-axis. And so symmetry is not required, but when it does exist, it's important to identify it.