Product of unbounded functions

Question

I am supposed to find out if a product of two unbounded functions can be a bounded function. Also if a composition of two unbounded functions can be a bounded function. Could anyone answer these questions and tell me why? Thanks

Answer 1

Let $f(x)=\frac 1x$

$g(x)=x$ and $h(x)=e^x$.

$f$ , $g$ and $h$ are unbounded at $(0,\infty)$

but $f(x)g(x)=1$ and $f(h(x))=e^{-x}$.

thus

$|(f.g)(x)|\le 1$ and $|f(h)(x)|\le 1$

hence, $fg$ and $foh$ are bounded at $(0,\infty)$.