Chapter 1 - Functions and Limits - 1.1 Four Ways to Represent a Function - 1.1 Exercises - Page 23: 80

if $f$ and $g$ are both even functions, $fg$ is even. if $f$ and $g$ are both odd functions, $fg$ is even. if $f$ is even and $g$ is odd, $fg$ is odd.

if $f$ and $g$ are both even functions, $(fg)(x) = f(x)g(x)$ $(fg)(-x) = f(-x)g(-x)=f(x)g(x)=(fg)(x)$ Therefore $fg$ is even. if $f$ and $g$ are both odd functions, $(fg)(x) = f(x)g(x)$ $(fg)(-x) = f(-x)g(-x)=(-f(x))(-g(x))=f(x)g(x)=(fg)(x)$ Therefore $fg$ is even. if $f$ is even and $g$ is odd, $(fg)(x) = f(x)g(x)$ $(fg)(-x) = f(-x)g(-x)=f(x)(-g(x))=-f(x)g(x)=-(fg)(x)$ Therefore $fg$ is odd.