Answer to Question #221668 in Discrete Mathematics for Sabelo Zwelakhe

Question

Let f: A$\to$ B and g: B$\to$ C be functions. Show that if g o f is onto, then g is onto.

Answer 1

 Suppose that g ◦ f is onto.

Let z ∈ C.

Then since g ◦ f is onto, 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 onto