trouble with bijection definition I have a bijection problem that I cannot get my head around. It goes like this: 

let f: A -> B and g: B -> C be functions such that g o f is a bijection. Prove that f must be one-to-one and that g must be onto. And also an example showing that it is possible for neither f nor g to be a bijection. 

This is the problem. I do not even know where to begin. Any help is appreciated.
Hints: First part: Suppose that $f$ is not one-to-one. That implies that $f(x) = f(y)$ for some $x \neq y$. Can $g \circ f$ be a bijection? Similar idea for the other half. Second part: We know that $f$ must be one-to-one and $g$ must be onto. So if $f$ and $g$ are not bijections, then $f$ must not be onto and $g$ must not be one-to-one. See how these two conditions mesh together.