 In this video, we're going to talk about what it means for a function, an algebraic function, to be symmetric. So a little bit of a definition. If a function f satisfies the relationship f of negative x is equal to f of x, then, and this happens for all x in the domain of the function, then we call this an even function, right? So f of negative x equals f of x means it's even. On the other hand, if the function satisfies a similar relationship, if f satisfies the relationship f of negative x is equal to negative f of x for every number x inside the domain, we call this an odd function. So let me try to explain one, why we have these names in the first place, and two, what's happening geometrically. So these right here are algebraic identities right here. So f of negative x equals negative f of x for odd functions, and f of negative x equals f of x for even functions. Now, geometrically, what's going on here? Let's let's first try to capture that idea. So let's let's consider the case that we have an even function f of negative x equals f of x. Well, when you look at the expression f of negative x for a second, what this means is this is a geometric transformation to the function. In fact, this is a reflect a reflection across the y axis. So what we're saying is if we have a function f and you take f of negative x, that wants you to reflect across the y axis. Well, if the function f of negative x is equal to f of x f of x is the original function. So what this is saying this algebraic sentence is telling us that if you reflect the graph across the y axis, nothing happens. You get something like maybe a parabola where reflection across the y axis doesn't change the graph reflection across the y axis is the same as the original function as well. And so an even function is exactly those those even function is one whose graph is symmetric with respect to the y axis and that's what we get as an even function for an odd function. This is the equation f of negative x equals negative f of x. And so like we saw a moment ago, this first one f of negative x, this is going to represent reflection across. Whoops. Across the y axis. On the other hand, if you take negative f of x there, this is geometrically another reflection, but this time it's reflection across the x axis. Now think about the for a moment with that means we're saying we want a graph which when you reflect across the y, it's the same thing as you reflect across the x axis. And geometrically, I'll let you try to figure this one out on your own here that if a function has the same reflection across the y axis as the reflection across the x axis, then it turns out this function is symmetric with respect to the origin. That is, the graph is left unchanged. When you rotate by pi radians or if you rotate by 180 degrees around the origin. And so this is an example of an odd function odd functions are symmetric with the origin if you take a half spin of the graph around the origin doesn't change the graph. Even functions are symmetric with respect to the y axis. That is, if you reflect it across the y axis the graph doesn't change. Let me show you some examples of this to kind of solidify the picture for us. So consider the following functions let's determine if they're even odd or neither because the function doesn't have to be even it doesn't have to be odd. So take the function f of x equals x to the fifth plus x. Now to algebraically test whether something's even or odd, what you're going to do is you're going to compute the value f of negative x, because if it's an even function f of negative x should equal f of x. If it's an odd function it should equal negative f of x. And if it's neither well you get something else. So the crux of this calculation is going to be look at f of negative x. So that means is everywhere in the expression where you see an x you're going to replace it with an negative x. So f of negative x you're going to take x to the fifth and replace with negative x to the fifth. Don't forget the parentheses that x will become a negative x. And now you're going to try to simplify this thing. One thing to note, when you have an odd power of a negative number that itself is going to be negative. In particular if you take negative one to the fifth power this is equal to negative one. So negative x raised to the fifth power is the same thing as just x to the fifth times negative one the negative one comes out of the power there. And of course negative x just becomes negative x you can drop the parentheses. And so notice this now looks like negative x to the fifth minus x. If we were to factor this negative sign out of the expression, we end up with negative one times x to the fifth plus x but wait a second, x to the fifth plus x that's just f of x itself. So f of negative x turns out to be negative f of x. So this is indicative that f is an odd function, which the graph of f is illustrated here on the right. And I want you to notice here that this graph is in fact symmetric with respect to the origin. If I were to take this graph and do a half spin, you're going to get the exact same graph again if you spin in 180 degrees, or if you take any point on the graph, and you draw through the origin the same distance on the side you'll find a point that's again symmetry with respect to the origin. Now, when we talk about odd functions right f of negative x is equal to negative x negative f of x. I want you to convince yourself that this reflection idea makes sense as well. If we were to reflect the graph across the y axis, we would get something that looks like the following. Like so, it's not an even function right. If you reflected it across the x axis though, you would end up with something that looks like the following. All right, and so notice this function which we claim to be odd has that geometric principle that reflection across the y axis is identical to a reflection across the x axis, which simply in a more simple way is just reflect rotation around the origin that's the symmetry we're talking about right here. Oh, and the other thing I want to mention here is why do we call these things odd functions? Well, in the case of a polynomial function, you'll notice the powers of x you have five and a one. A polynomial is an odd function if and only if all the powers of x are odd. That is, we're not counting the ones with zero coefficients, of course, only those powers of x that are present. That's actually why we call them odd functions because for polynomials, it's very easy to detect all the powers are going to be odd. Now that's not true for general functions. You can have, for example, a rational function, all whose powers of x are odd top and bottom, but the function would still be even. So the best thing to do is to check by looking at things like g of negative x. But for polynomials, it's a little bit easier. So let's take g of x this time to be one minus x to the fourth. If we want to test for symmetry, we're going to look at g of negative x. For which everywhere we see an x, we're going to replace it with a negative x. So we get one minus negative x to the fourth. Now, when you take negative one to an even power, you're always going to get back positive one. And therefore, negative x to the fourth becomes just an x to the fourth. In which case, notice you'll have one minus x to the fourth, one minus x to the fourth. That's just g again. That should be g not f right there. And so you get g of negative x equals g of x and thus g is an even function. Whoops a daisy. Because we get g of negative x equals to g of x right here. In terms of exponences, this is a polynomial function, right? You have one minus x to the fourth, you have an even power right there. One could be written as x to the zero, which zero is an even power right there. So the same thing is true for polynomials. If all of the powers of x are even, it'll be an even function. Look at the graph of g over here. This function is in fact symmetric with respect to the y-axis. If you reflect it across the y-axis, you get the exact same graph again. Let's put that back on there. And then one last example here. Let's consider h of x this time to equal 2x minus x squared. If you test for symmetry, we're going to look at h of negative x for which you replace the x with negative x and then the x squared becomes a negative x squared. Now for odd powers of x, right, the negative sign will stick around. So this becomes a negative 2x. But for even powers, if you have negative x squared, that's going to become a positive x squared. So you're going to get negative 2x minus x squared. I'll point this out to you that this is not the same thing as h of x. So we know this thing is not going to be an even function. But if we try to factor out the negative sign, if you factor out the negative sign, you get negative 1 times 2x plus x squared. This is not the same thing as negative h of x either. So it's not an odd function. So we actually would conclude that this function h is neither even nor odd. And when you look at the graph here, you don't see either of those type of symmetries. It's not symmetric with respect to the y-axis. It's not symmetric with respect to the origin. Now I should mention that the graph is a parabola. It does have an axis of symmetry. And so this graph would be symmetric with respect to x equals 1. But when we look for symmetry, we're going to look for symmetry with respect to 0. Because the y-axis is the line x equals 0 and the origin is the point 00. So when we ask for symmetry of a function, we're really interested in, is it symmetric with respect to 0? Is it symmetric with respect to y-axis being even or symmetric with respect to the origin being odd?