Question

In asking whether or not the function f(x)=(1+x)^4 is even or odd, wouldn't it be neither? Seeing that f(-x)=(1-x)^4 is neither f(x) nor -f(x)?

Answer 1

Yes, f(x) is neither even nor odd.
However, you can break it into an even and an odd function that sum to f(x):
\[ f_e(x)= \frac{f(x)+f(-x)}{2} \\ f_o(x)= \frac{f(x)-f(-x)}{2} \]
Here is a graph of f(x) and the even (red) and odd (blue) that add to f(x) shown in green and the sum shown in orange.

Answer 2

Great, thank you. Yeah, I suppose one of the problems I was having is that in the problems it lists the equation as \[f_e(x) +f_o(x) = \frac{ f(x)+f(-x) }{ 2 } + \frac{f(x)+f(-x)}{2}\]
It makes a lot most sense with the second fraction having a minus sign.