This is a question I've had a tough time getting a good answer to.
Consider the problem to minimize $f(x)$. Assume $f$ is differentiable and nice in every way, but we do not know if $f$ is convex.
A common approach is to instead minimize $g(y)$ where $y=f(x)$. Oftentimes, $g(y)$ will be convex and we are assured that any local minimum is a global minimum, call it $x^*$. But it is well known that convexity of $g(f(x))$ does NOT imply convexity of $f(x)$. So how do we know that $x^*$ globally minimizes $f(x)$?
If $g$ is monotone and $g(f(x))$ is convex, then $f(x)$ is always convex, right? But does this imply that all non-monotonic manipulations of $f$ are 'disallowed'?
$\exp(\log(x))$ is convex, $\exp$ is monotone, but $\log(x)$ is not convex.
Convex optimization by Boyd and Vandenberghe would be a suitable introductory text on compositions of convex functions etc.