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.