I am trying to give an explination or proof as to why F(G(X)) is always an even function if F(X) is even and G(X) is either even or odd. I understand why in simple situations such as when the resulting function is (X^3)^2 or (X^2)^2. It's because the product of two even numbers OR an odd number and a even number are both even.

But what if the resulting function is something more complicated such as
(X^3+X^5)^2. At first I thought that the outside function determines the symmetry, but then I tried
(X^2+X^4)^3, where the outside function is odd, and realized that this function is also even. I am stumped on how to prove, or even put into words, the reason for this function behavior.

Use the definitions of even and odd functions. In this case, viewing details yields little insight.
In other words, simply find the sign of F(G(-x)) vs. F(G(x)) using only the defining behavior of even and odd functions.

cristo
Staff Emeritus
What are the properties of odd and even functions, in general?

Wow, super fast post by slider!

think about what it means for a function to be even, by definition of even function.

A function f is even if f(x)=f(-x) for all x.

Consider f(g(-x)).
we have two cases, g is even or g is odd, we will look at each individually.

if g is even then f(g(-x)) = f(g(x)) so f(g(x)) is even by definition of even.

Consider the case where g is odd (i.e. g(-x) = -g(x)). If g is odd we know (f(g(-x)) = f(-g(x)). but we know f is even so f(-x)=f(x) FOR ANY X (including g(x)). so f(-g(x)) = f(g(x)). But this means f(g(-x)) = f(g(x)) thus f(g(x)) is even.

cristo
Staff Emeritus
Or you could just tell him the answer! This is probably a homework question (if not, it sure looks like one) but even if it's not, it's better to let someone think about the question and the definitions of the functions, than to simply tell him what to do!

Use the definitions of even and odd functions. In this case, viewing details yields little insight.
In other words, simply find the sign of F(G(-x)) vs. F(G(x)) using only the defining behavior of even and odd functions.

In an even function all points are reflected across the Y axis. In other words for every point (X,Y) there is a corresponding point (-X,Y). The value of Y does not change when X is replaced by -X, i.e. F(-x)= F(X)

An odd function has an opposite value for Y when X is replaced by -x, i.e.
G(-x) = -G(x).
--------------------------------------
So here are the most basic properties of even and odd functions,F is even and G is odd:

F(-X) = F(X)

G(-X)=-G(X)

F(G(X)) = X

F(G(-X)) = X

Is this what you mean?

think about what it means for a function to be even, by definition of even function.

A function f is even if f(x)=f(-x) for all x.

Consider f(g(-x)).
we have two cases, g is even or g is odd, we will look at each individually.

if g is even then f(g(-x)) = f(g(x)) so f(g(x)) is even by definition of even.

Consider the case where g is odd (i.e. g(-x) = -g(x)). If g is odd we know (f(g(-x)) = f(-g(x)). but we know f is even so f(-x)=f(x) FOR ANY X (including g(x)). so f(-g(x)) = f(g(x)). But this means f(g(-x)) = f(g(x)) thus f(g(x)) is even.
Thanks Jon that makes sense. So the first step is to realize that an opposite value of X on the inside odd function will yield an opposite value for that whole function G(X), then mentally replace -G(X) with -X and remember that for an even function F(-X)=F(X). That is so cool.

In an even function all points are reflected across the Y axis. In other words for every point (X,Y) there is a corresponding point (-X,Y). The value of Y does not change when X is replaced by -X, i.e. F(-x)= F(X)

An odd function has an opposite value for Y when X is replaced by -x, i.e.
G(-x) = -G(x).
--------------------------------------
So here are the most basic properties of even and odd functions,F is even and G is odd:

F(-X) = F(X)

G(-X)=-G(X)
Up to that part, it's good.

F(G(X)) = X
This is only true if F is the inverse function of G. We don't know anything specific about F(G(X)) except that F is even and G is odd. But using the definitions we know, we can simplify the expression F(G(-X)), as Jon has done.