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Master Polynomial expressions, equations, and functions with these 15 lessons for Algebra II classes.
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Let's see if we can learn a thing or two about even functions and odd functions. So even functions, and on the right-hand side over here, we'll talk about odd functions. If we have time, we'll talk about functions that are neither even nor odd. So before I go into kind of a formal definition of even functions, I just want to show you what they look like visually, because I think that's probably the easiest way to recognize them. And then it'll also make a little bit more sense when we talk about the formal definition of an even function. So let me draw some coordinate axes here, x-axis. And then, let me see if I can draw that a little bit straighter. This right over here is my y-axis, or I could say y is equal to f of x axis, just like that. And then let me draw the graph of f of x. f of x is equal to x-squared, or y is equal to x-squared, either one. So let me draw it. In the first quadrant, it looks like this. And then in the second quadrant, it looks like this. It looks like-- oh, let me try to draw it so it's symmetric. Pretty good job. The f of x is equal to x squared is an even function. And the way that you recognize it is because it has this symmetry around the y-axis. If you take what's going on on the right-hand side, to the right of the y-axis, and you just reflect it over the y-axis, you get the other side of the function. And that's what tells you it is an even function. And I want to show you one interesting property here. If you take any x-value-- let's say you take a positive x-value. Let's say you take the value, x is equal to 2. If you find f of 2, that's going to be 4. That's going to be 4 for this particular function for f of x is equal to x squared. 2 squared is equal to 4. And if you took the negative version of 2-- so if you took negative 2, and you evaluated the function there, you are also going to get 4. And this, hopefully, or maybe makes complete sense to you. You're like, well, Sal, obviously if I just reflect this function over the y-axis, that's going to be the case. Whatever function value I get at the positive value of a number, I'm going to get the same function value at the negative value. And this is what kind of leads us to the formal definition. If a function is even-- or I could say a function is even if and only of-- so it's even. And don't get confused between the term even function and the term even number. They're completely different kind of ideas. So there's not, at least an obvious connection that I know of, between even functions and even numbers or odd functions and odd numbers. So you're an even function if and only if, f of x is equal to f of negative x. And the reason why I didn't introduce this from the beginning-- because this is really the definition of an even function-- is when you look at this, you're like hey, what does this mean? f of x is equal to f of negative x. And all it does mean is this. It means that if I were to take f of 2, f of 2 is 4. So let me show you with a particular case. f of 2 is equal to f of negative 2. And this particular case for f of x is equal to x squared, they are both equal to 4. So really, this is just another way of saying that the function can be reflected, or the left side of the function is the reflection of the right side of the function across the vertical axis, across the y-axis. Now just to make sure we have a decent understanding here, let me draw a few more even functions. And I'm going to draw some fairly wacky things just so you really kind of learn to visually recognize them. So let's say a function like this, it maybe jumps up to here, and maybe it does something like that. And then on this side, it does the same thing. It's the reflection, so it jumps up here, then it goes like this, then it goes like this. I'm trying to draw it so it's the mirror image of each other. This is an even function. You take what's going on on the right hand side of this function and you literally just reflect it over the y-axis, and you get the left hand side of the function. And you could see that even this holds. If I take some value-- let's say that this value right here is, I don't know, 3. And let's say that f of 3 over here is equal to, let's say, that that is 5. So this is 5. We see that f of negative 3 is also going to be equal to 5. And that's what our definition of an even function told us. And I can draw, let me just draw one more to really make sure. I'll do the axis in that same green color. Let me do one more like this. And you could have maybe some type of trigonometric looking function that looks like this, that looks like that. And it keeps going in either direction. So something like this would also be even. So all of these are even functions. Now, you are probably thinking, well, what is an odd function? And let me draw an odd function for you. So let me draw the axis once again. x-axis, y-axis, or the f of x-axis. And to show you an odd function, I'll give you a particular odd function, maybe the most famous of the odd functions. This is probably the most famous of the even functions. And it is f of x-- although there are probably other contenders for most famous odd function. f of x is equal to x to the third power. And it looks like-- you might have seen the graph of it. If you haven't, you can graph it by trying some points. It looks like that. And the way to visually recognize an odd function is you look at what's going on to the right of the y-axis. Once again, this is the y-axis, this is the x-axis. You have all of this business to the right of the y-axis. If you reflect it over the y-axis, you would get something like this. And if the left side of this graph looked like this, then we would be dealing with an even function. Clearly it doesn't. To make this an odd function, we reflect it once over the y-axis and then reflect it over the x-axis. Or another way to think about it, reflect it once over the y-axis and then make it negative. Either way, it will get you there. Or you could even reflect it over the x-axis and then the y-axis, so you are kind of doing two reflections. And so clearly if you take this up here and then you reflect it over the x-axis, you get these values, you get this part of the graph right over here. And if you try to do it with a particular point, and I'm doing this to kind of hint at what the definition, the formal definition of an odd function is going to be. Let's try a point, let's try 2 again. If you had the point 2, f of 2 is 8. So f of 2 is equal to 8. Now what happens if we take negative 2? f of negative 2, negative 2 to the third power, that's just going to be negative 8. So f of negative 2 is equal to negative 8. And in general, if we take-- so let me just write it over here. f of 2-- so we're just doing one particular example from this particular function. We have f of 2 is equal to, not f of negative 2. 8 does not equal negative 8. 8 is equal to the negative of negative 8 because that's positive 8. So f of 2 is equal to the negative of f of negative 2. We figured out-- just want to make it clear-- we figured out f of 2 is 8. 2 to the third power is 8. We know that f of negative 2 is negative 8. Negative 2 to the third power is negative 8. So you have the negative of negative 8, negatives cancel out, and it works out. So in general, you have an odd function. So here's the definition. You are dealing with an odd function if and only if f of x for all the x's that are defined on that function, or for which that function is defined, if f of x is equal to the negative of f of negative x. Or you'll sometimes see it the other way if you multiply both sides of this equation by negative 1, you would get negative f of x is equal to f of negative x. And sometimes you'll see it where it's swapped around where they'll say f of negative x is equal to-- let me write that careful-- is equal to negative f of x. I just swapped these two sides. So let me just draw you some more odd functions. So I'll do these visually. So let me draw that a little bit cleaner. So if you have maybe the function does something wacky like this on the right hand side. If it was even, you would reflect it there. But we want to have an odd function, so we're going to reflect it again. So the rest of the function is going to look like this. So what I've drawn in the non-dotted lines, this right here is an odd function. And you could even look at the definition. If you take some value, a, and then you take f of a, which would put you up here. This right here would be f of a. If you take the negative value of that, if you took negative a here, f of negative a is going to be down here. So f of negative a, it's going to be the same distance from the horizontal axis. It's not completely clear the way I drew it just now. So it's maybe going to be like right over here. So this right over here is going to be f of negative a, which is the same distance from the origin as f of a, it's just the negative. I didn't completely draw it to scale. Let me draw one more of these odd functions. I think you might get the point. Actually, I'll draw a very simple odd function, just to show you that it doesn't always have to be something crazy. So a very simple odd function would be y is equal to x, something like this. Whoops. y is equal going through the origin. You reflect what's on the right onto to the left. You get that. And then you reflect it down, you get all of this stuff in the third quadrant. So this is also an odd function. Now, I want to leave you with a few things that are not odd functions and that sometimes might be confused to be odd functions. So you might have something like this where maybe you have a parabola, but it's not symmetric around the y-axis. And your temptation might be, hey, there is this symmetry for this parabola. But it's not being reflected around the y-axis. You don't have a situation here where f of x is equal to f of negative x. So this is neither odd nor even. Similarly, you might see, let's say you see a shifted cubic function. So say you have something like this. Let's say you have x to the third plus 1. So f of x is equal to x to the third plus 1, so it might look something like this. And once again, you will be tempted to call this an odd function. But because it's shifted up, it is no longer an odd function. You can look at that visually. So this is f of x is equal to x to the third plus 1. If you take what's on the right hand side and reflect it onto to the left hand side, you would get something like that. And then if you were to reflect that down, you would get something like that. So this is not an odd function. This isn't the left reflection and then the top-bottom reflection of what's going on on the right hand side. This over here actually would be.