Prove that $g$ is injective or surjective

Question

I have two questions which are related to mappings as follow:

Given 3 sets E, F, and G such that $f: E \rightarrow F, g: F \rightarrow G$ are two mappings. Prove that:

a. If $g \circ f$ is injective and $f$ is surjective then $g$ is injective.

b. If $g \circ f$ is surjective and $g$ is injective then $f$ is surjective.

For these two problems. I have solved halfway for each.

For the problem (a), I have proved that $f$ is bijective.

For the problem (b), I have proved that $g$ is bijective.

I don't know how to do next because I don't find the relation to conclude.

Answer 1

In part a, because $g \circ f$ is injective and $g^{-1}$ (which exists, since $g$ is bijective) is injective, we therefore have $$f = g^{-1} \circ (g \circ f)$$ is the composition of injective functions, which hence makes it injective. Part b can be solved dually.