If $f,g,h$ are functions defined on $\mathbb{R},$ the function $(f\circ g \circ h)$ is bounded:
i) If $f$ is bounded.
ii) If $g$ is bounded.
iii) If $h$ is bounded.
iv) Only if all the functions $f,g,h$ are bounded.
Shall I take different examples of functions and see?
Could anyone explain this for me please?
As $\Im f\circ g\circ h\subseteq\Im f$, the boundedness of $f\circ g\circ h$ is controlled by $f$.
For ii), let $f(x)=1/x$, $g(x)=h(x)=x$, $x\in(0,1)$, both $g$ and $h$ are bounded, but $f\circ g\circ h=f$ is still unbounded.
The rest should be similar.