If a convex function is divided by a linear function, what will it be?

Question

I have a convex function $f(x)$ and a linear function $g(x)$. Is $\frac{f(x)}{g(x)}$ still convex? If it is not, can I use the techniques in convex optimization to optimize $\frac{f(x)}{g(x)}$?

Answer 1

$f(x)=x^4$ is a convex function, and $g(x)=x$ is linear, but $\frac{f(x)}{g(x)} = x^3$ which is not convex