Function composition with differing join ends

Question

If $f:A\to B$ and $g:C\to D $ are two functions. When does $g\circ f$ make sense?

My thinking: when $\operatorname{Im}(f)\subseteq C$.

Answer 1

You are right. The function $g\circ f$ makes sense when and only when $f(A)\subset C$.

Answer 2

Of course when any part of the image of $f$ is not within $C$ the mapping will not be possible.

So $f(A)\subseteq C$ is necessary for $g\circ f$ to be sensible.

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Notice, this is not the same thing as requiring $B\subseteq C$, as the image may not be the entirety of $B$.