Even/odd functions & numbers

in the last video and even and odd functions I talked about how you shouldn't get confused between even functions and even numbers and odd functions and odd numbers and I said there wasn't any obvious connection between the word even function and the our notion of even numbers or any connection between odd functions and odd numbers and I was wrong there actually is a relatively obvious connection this was pointed out by the YouTube user know theists and the connection and it almost I almost kind of explicitly did it in the last example when I showed it an even function I showed you x squared when I showed you an odd function I showed you X to the third power when I wanted to show you another odd function I showed you Y is equal to X or f of X is equal to X to the first power and so you might start to notice what no theists pointed out is that kind of these these archetypical or I guess you know these good examples or these simple examples of even and odd functions when I'm just when I just have a very simple X raised to some power the whether the power is even or odd does describe or it's it's it's going to tell you whether the function is even or odd and you want to be very careful here not all not all even or odd functions even have exponents in them they could be trigonometric functions there might be other some other type of wacky functions you don't have to have exponents it's just that these exponents are probably where the motivations for calling these even functions and odd functions came from and let me just be clear it's not just also any polynomial and even in the last video when we had X to the third plus one this was neither even or odd but if you just have the pure X raised to some power then all of a sudden it because the the notions of even and odd seem to or I get some motivations for calling even and odd so you start to make sense because if I have f of X is equal to X to the first power that's the same thing as Y is equal to X this is odd and that and it kind of gels with the name because we are also raising it to an odd power if we have f of X is equal to x squared we saw in the previous video this is even and it kind of gels with the idea that we're raising it to an even power I could keep going if it was to the X 2/3 that is odd I could keep going if we're raising it to any if we just have let me write it this way in general if you have f of X is equal to X to the N then this is odd odd function if n is odd is an odd number and this is an even function even function if n is even and I want to make it very clear here those whole the whole point of this video is just to clarify the motivation for calling them even or odd functions not all even functions are going to be of of this form here where it's X raised to some even power and not all odd functions are going to be and I also don't want you to be confused that if I have something like X to the third and then I have other stuff Oh past that you say oh X to the third that's an odd number but this is not an odd function just when it's just kind of a pure down a pure stripped down X to the third or X to the first can you really make that statement but that really is probably where the motivation comes for naming them even or odd functions and then the other symmetric functions even if they even if they don't let's say the other even functions even if they don't involve an exponent maybe this is some type of trigonometric function it's kind of you're calling it even because you're saying well it's kind of like it has the same type of symmetry as a as a as say x squared or x to an even power so you kind of group them all together as even functions and then all of these even though this may or may not have an exponent in it it has the same type of symmetries as X raised to an odd power so that's why we call them odd functions well thank you know theist for that that that that pointing that out