Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 66

If $g$ is an odd function, let $h = f \circ g$. Is $h$ always an odd function? What if $f$ is odd? What if $f$ is even?


$
\begin{equation}
\begin{aligned}

\text{Let } g(x) =& x^3\\
f_1(x) =& x^5\\
f_2(x) =& x^6\\
\\
h_1 =& f \circ g\\
h_1 =& (x^3)^5\\
h_1 =& x^{15}\\
\\
h_2 =& f \circ g\\
h_2 =& (x^3)^{16}\\
h_2 =& x^{18}

\end{aligned}
\end{equation}
$



It's unpredictable, we must know first the function $f$ ; if $f$ is odd, $h$ will be odd also. On the other hand, if $f$ is even, $h$ will be even as well.