 This video is going to talk about odd and even functions and even function is defined as f of negative x is equal to f of x And what that really means is if it's an even function you are going to have the point x y on my graph I'm also going to have the point negative x y. I have a negative x here and the same That's a positive y output. Let's look at the points first I have the point 1 0 here and on my graph I should have negative 1 0 when I change the x sign and I do now I look at my graph And I'll talk about symmetry. Do you look at this y-axis if I were to put a mirror on this y-axis? Then if I looked at this side of my graph in my mirror would show up this side of my graph their mirror images or Symmetrical about the y-axis and that's going to be important. It has to be about the y-axis I could have a graph that looked just like this But if it moved over it wouldn't be an even function Because it would have caused me to not be symmetrical about the y-axis So this is very important just because it's an even exponent doesn't mean it's an even function Looking algebraically we have our f of x up there as x to the fourth plus 2 x squared minus 3 If I put in negative x when I get done I should get back to this exact same thing But remember this means put a negative x and then to the fourth plus 2 times negative x squared Minus 3 negatives to an even power will be positive So that's x to the fourth plus 2 x squared minus 3 which is my f of x one other thing that we can talk about and that is to talk about the exponents if You notice all exponents are equal to an even And even if you think about this one it would be x to the zero and a zero could be positive or negative So it also is even in this case. So you have symmetrical about the y-axis You have opposite signs on your x's and you can prove that by just plugging in negative x to your function Getting the function back out or you can look at the function and see or all my exponents even then it's going to be an Even function so let's move to odd functions odd functions say that if I put in negative x I'm going to get the opposite of f of x or if I have an x y I'm going to have a negative x negative y So if I have one and one point five I should have what that means is negative one and negative one point five Let's look and see if we do here's my looks like one and one point five and negative one down to negative one point five Sure enough there it is on my graph. So let's talk about the symmetry the symmetry goes this way It's about the origin. Those are a little bit harder to see or region But if you take your pencil on your paper and put it right there on the origin and then turn your paper 180 degrees you're going to end up with this exact same graph. All right So the last thing we had to do then is look at it algebraically So we have our f of negative x this time if we put in negative x we should get in the exact opposite Function so we have negative x being cubed plus point five Times negative x or this is a negative being cubed which will be a negative So negative x cubed point five times a negative x will give me a negative point five x and you notice that this is a negative And a negative both my signs are negative and in my original problem both my signs were positive So the signs are opposite So I know that it is going to be an odd function and again. Let's look at our exponents. Does the rule still apply? Yes, it does all these exponents are equal to an odd number. That is odd and even functions