Kenneth Rosen Edition 7 Exercise 2.3 Question 35 (Page No. 154)

Question

If $f$ and $fog$ are onto, does it follow that $g$ is onto?Justify your answer.

Answer 1

Suppose that g ◦ f is surjective. Let z ∈ C. Then since g ◦ f is surjective, there exists x ∈ A such that (g ◦ f)(x) = g(f(x)) = z. Therefore if we let y = f(x) ∈ B, then g(y) = z. Thus g is surjective.