Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 63

(a) If $f$ and $g$ are even functions. What can you say about $f+g$ and $fg$?


$\begin{equation} \begin{aligned} \text{Let } f(x) =& x^2\\ g(x) =& x^4\\ \\ f + g =& x^2 + x^4\\ fg =& x^2(x^4)\\ fg =& x^6 \end{aligned} \end{equation}$


Therefore, $f + g$ and $fg$ are even functions.

(b) Then what if $f$ and $g$ are both odd?


$\begin{equation} \begin{aligned} \text{Let } f(x) =& x^3\\ g(x) =& x^5\\ \\ f + g =& x^3 + x^5\\ fg =& x^3 (x^5)\\ fg =& x^8 \end{aligned} \end{equation}$


Therefore, $f + g$ is an odd function while $fg$ is even.