Hello,

I'm supposed to find the even and odd parts of the following $$f(x)=\begin{cases}x^2 & x<0 \\ e^{-x} & x>0.\end{cases}$$

We have a formula, $f(x)=f_e(x)+f_o(x)=1/2 (f(x)+f(-x))+1/2(f(x)-f(-x))$.

So for even/odd to mean anything $x$ should live in an interval which is symmetric with respect to 0. For otherwise there would exist some $x_0$ such that either $f(x)\neq f(-x)$ or $f(-x) \neq -f(x)$ simply because the function isn't defined on appropriate values.

I tried just naively shoving function values into the formula. So $f_e(x)=1/2 (f(x)+f(-x))=(e^{-x}+x^2)$ but since $f(-x)=1/2(e^{-(-x)}+(x)^2)$ this function is not even. So this can't be correct.

Is there some mistake?