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.