For function composition, can we take a subset of the codomain of the inner function as the domain of the outer function?

Let $f:A \rightarrow B$ and $g:B \rightarrow C$ be functions. Suppose that $f$ is not surjective. I want to construct a function composition $g\circ f$. But because there is at least one $b \in B$ for every $a \in A$ such that $b\neq f (a)$, it follows that $g$ is not defined for every $b \in B$ insofar as we cannot then construct the composition

$$g \circ f : A \rightarrow C \mathrm {\, \, \, \, defined \, \,by \, \, \, \, } (g\circ f)(x)= g (f (x))$$

However, is it permissible to take the image $f (A) \subset B$ as the domain of $g$? That is, $g: f (A) \rightarrow C$. Then we are guaranteed that every $b=f (a) \in f (A)$ is mapped by $g$ to a unique element in $C$, that is, $g\circ f$ is well-defined.

Intuitively, this makes sense, but I am not sure if it is permissible. I hope it is clear what I am asking.