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.