# Composite function is bijective

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?

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.)
• No g must be surjective for the composition to be surjective. I know this proof! And f is injective for the composition to be injective. So would that mean that for this case g is both surjective and injective hence bijective? – Molly Mar 20 at 17:42