About this video
Sal corrects his earlier claim and explains the link between even/odd functions and even/odd numbers: for a pure power f(x) = x^n, the function is even when n is even and odd when n is odd. He warns that this only works for a single pure power, and that other functions (like trig functions) are called even or odd because they share the same symmetry.
You'll be able to
- say whether f(x) = x^n is even or odd from the exponent n.
- explain why the names 'even function' and 'odd function' were chosen.
- recognise that adding other terms (like x^3 + 1) can make a function neither even nor odd.
Key words
- even function
- A function with the same symmetry as x^2: f(-x) = f(x), symmetric about the y-axis.
- odd function
- A function with the same symmetry as x^3: f(-x) = -f(x).
- exponent
- The power x is raised to, like the 3 in x^3.
- symmetry
- When one part of a graph is a mirror or rotated copy of another part.
Transcript
In the last video on even and odd functions, I talk about how you shouldn't get confused between even functions and even numbers and odd functions and odd numbers. And I said that there wasn't any obvious connection between the word even function and 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, and this was pointed out by the YouTube user Nothias. And the connection, I almost explicitly did it in the last example. When I showed 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 Nothias pointed out, is that these archetypal or these good examples or these simple examples of even and odd functions, when I just have a very simple x raised to some power, whether the power is even or odd, it's going to tell you whether the function is even or odd.
And you want to be very careful here. Not all even or odd functions even have exponents in them. They could be trigonometric functions. They might be 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 1, this was neither even or odd-- but if you just have the pure x raised to some power, then all of the sudden, the motivations for calling them even and odd 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 it 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 gels with the idea that we're raising it to an even power. I could keep going. If it was to the x to the third, that is odd.
I could keep going. 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, if n is even. And I want to make it very clear here. 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 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 past that, and you say, oh, x to the third, that's an odd number.
But this is not an odd function. Just when it's just 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 don't involve an exponent-- maybe this is some type of trigonometric function-- you're calling it even because you're saying it has the same type of symmetry as, say, x squared or x to an even power.
So you 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 call them odd functions. Well, thank you Nothias for pointing that out.
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