$f$ is convex or concave?

Question

If $ f(x)$ be a concave function, what can be said about the convexity of $f(x)=1/g(x)$

I thinking don't have convexity for $1/g(x)$ because maybe $g(x)=0$ , $g(x)\leq 0$

But I have no idea for proof, my guess is true? how can Begin the proof?