These are all in the same vein so I figured it would be acceptable to post all-together.

Suppose that go f is bijective. Does it follow that g is injective? Prove your claim. (Hint: If your answer is yes, show that g is indeed injective; if your answer is no, present a pair of functions f and g as a counterexample.) Suppose that gof is bijective. Does it follow that f is injective? Prove your claim Suppose that gof is bijective. Does it follow that g is surjective? Prove your claim Suppose that gof is bijective. Does it follow that f is surjective? Prove your claim