Let $g : A \to B$ and $f : B \to C$ be functions. Let $f \circ g$ be onto. Are both $f$ and $g$ necessarily onto?

Question

What would the steps be for someone new trying to learn how to solve this? I feel like I understand the concepts and what they mean but not how to prove it. Thanks

Answer 1

The function $f$ must be onto, as it maps $g[A]$ onto $C$. Though, $g$ need not be onto. Consider the function $g:\{1\}\to\{0,1\}$ defined by $g(1)=1$, and function $f:\{0,1\}\to\{1\}$ defined by $f(0)=f(1)=1$. It follows that $f\circ g(1)=1$ so $f\circ g$ is onto. However, $g[\{1\}]=\{1\}\not= \{0,1\}$