Composite function is bijective

Question

Suppose $f : X → Y$ and $g : Y → Z$ are functions. If $g ◦ f$ is bijective and $f$ is surjective. Then what would $g$ be? Would it be bijective or invective? I know that $g ◦ f$ is injective then $f$ is injective and that if $g ◦ f$ is surjective then $g$ is surjective. And I know how to prove these but how would I show if $g ◦ f$ is bijective and if $f$ is surjective what $g$ would be?

Answer 1

Hint:

• If $g$ isn't surjective, can $g\circ f$ be surjective?
• If $g$ isn't injective, can $g\circ f$ be injective? (Here you need to use that $f$ it's surjective.)