We have just shown that a composition of injective (resp. surjective, resp. bijective) functions are again injective (resp. surjective resp. bijective). In general, we cannot presume that the converse of these statements is true. However, the following statements can be proven:
Let $f:A\mapsto B$ and $g:B\mapsto C$ be functions. 1. If the composition $g\circ f$ is injective, then $f$ is injective. 1. If the composition $g\circ f$ is surjective, then $g$ is surjective. 1. If $f$ is surjective and $g\circ f$ is injective, then $g$ is injective. 1. If $f$ is surjective and $g\circ f$ is injective, then $g$ is injective.
Proofs: 1
