Answer
$h$ is an odd function if $f$ is an odd function.
$h$ is an even function if $f$ is an even function.
Therefore, $h$ is not always an odd function.
Work Step by Step
It is given that $g(x)$ is an odd function.
Then $g(-x) = -g(x)$
Suppose that $f(x)$ is an odd function.
Then $f(-x) = -f(x)$
We can consider $h(-x)$:
$h(-x) = f \circ g(-x)$
$= f \circ -g(x)$
$= -[f \circ g(x)]$
$= -h(x)$
$h$ is an odd function if $f$ is an odd function.
Suppose that $f(x)$ is an even function.
Then $f(-x) = f(x)$
We can consider $h(-x)$:
$h(-x) = f \circ g(-x)$
$= f \circ -g(x)$
$= f \circ g(x)$
$= h(x)$
$h$ is an even function if $f$ is an even function.
Therefore, $h$ is not always an odd function.
