(1) If $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are convex functions, then $f\circ g$ is convex. True or false? If not give a counter-example.
(2) If $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ are concave functions, then $f\circ g$ is concave. True or false? If not give a counter-example.
Clearly both statements are false. I computed a counter-example for each of them and I would like to know if they hold, please.
(1) Let $g(x)=x^2-1$ and $f(x)=|x|$. Both $f(x)$ and $g(x)$ are convex on $\mathbb{R}$. But, $f\circ g (x)=|x^2-1|$ is not convex on $\mathbb{R}$. So, the statement (1) doesn't hold
(2) Let $g(x)=-e^x$ and $f(x)=-x$. Both functions are concave on $\mathbb{R}$, but $f\circ g(x)=e^x$ is convex. So the statement (2) doesn't hold.
