Take that $g(x)$ is continuous and surjective, and that the codomain of $f(x)$ is a metric space. Is this enough?
One knows that $(f(g(W)))^{-1}=g^{-1}(f^{-1}(W))$ is open if $W$ is open in the codomain. We also know that $g^{-1}(V)$ is open for $V$ an open set. However, $g^{-1}(Z)$ can still be open if $Z=f^{-1}(W)$ is not open, right? Are there additional conditions on $g$ that could be put in place to allow for the result? Do we already have enough?
ASSUME $g$ is not constant.