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.