# Math Help - onto and composition proof

1. ## onto and composition proof

I've got this review problem:

Let f: X -> Y and g: Y -> Z be functions such that gf: X -> Z is onto. Prove that g must be onto.
When I draw it out on paper, it seems quite intuitive: if g is not onto, then there is a "connection" that can't be made between X and some member of Z. But I'm having trouble thinking of how to formally express that...

2. ## Re: onto and composition proof

Originally Posted by infraRed
I've got this review problem:

Let f: X -> Y and g: Y -> Z be functions such that gf: X -> Z is onto. Prove that g must be onto.

When I draw it out on paper, it seems quite intuitive: if g is not onto, then there is a "connection" that can't be made between X and some member of Z. But I'm having trouble thinking of how to formally express that...
Suppose that g is not onto. Then $\exists z_0 \in Z \ni g(y) \neq z_0~\forall y \in Y$
Does $gf(x) = z_0$ for some $x \in X$ ? Can $gf(\cdot)$ thus be onto?