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# Even/odd functions & numbers

Video 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.