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## Main Question or Discussion Point

I am trying to give an explination or proof as to

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.

__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.