This is a question I have on a test paper:

(I wrote "g o f" when I mean the "composite function of g and f").Is the following statement always true? Justify your answer.

If $\displaystyle f:A \rightarrow B$ and $\displaystyle g:C \rightarrow D$ are surjections where $\displaystyle Im(f) \subset C$, then g o f: $\displaystyle A\rightarrow D$ is a surjection.

I would say this statement is true because the domain of g is contained in the image of f (which is a necessary criterion for g o f to exist).

The composite function of two surjections is also a surjection.

Both these facts mean that the statement is always true.

I think this is right, I just wanted to see if I had overlooked anything.