Answer
$(fg)(-x)=(fg)(x)$ if both $f(x)$ and $g(x)$ are odd or if both $f(x)$ and $g(x)$ are even.
Work Step by Step
1) Let $f(x)$ and $g(x)$ be two odd functions. That is:
$f(-x)=-f(x)$ and $g(-x)=-g(x)$
$(fg)(x)=f(x)g(x)$
So,
$(fg)(-x)=f(-x)g(-x)=-f(x)[-g(x)]=f(x)g(x)=(fg)(x)$
2) Let $f(x)$ and $g(x)$ be two even functions. That is:
$f(-x)=f(x)$ and $g(-x)=g(x)$
$(fg)(x)=f(x)g(x)$
So,
$(fg)(-x)=f(-x)g(-x)=f(x)g(x)=(fg)(x)$
