Let f(x) and g(x) be increasing and decreasing functions respectively from $[0, \infty)$ to $[0, \infty)$. Let h(x) = fog(x). If h(0) = 0, then h(x), is

always 0

Since composition of an increasing function and a decreasing function is always a decreasing function. Therefore, $h(x):[0, \infty) \rightarrow[0, \infty)$ is a decreasing function.

Hence,

$h(x) \leq h(0)$ for all $x \geq 0$

$\Rightarrow h(x) \leq 0$ for all $x \geq 0$

$\Rightarrow h(x)=0$ for all $x \geq 0$        $\left[\begin{array}{l}∵ h(x) \in[0, \infty) \\ \Rightarrow h(x) \geq 0 \text { for all } x \in[0, \infty)\end{array}\right]$