If $g\circ f$ is one-to-one, is it true that $f$ is onto or $g$ is onto?

Question

Let $S,T$ and $U$ be nonempty sets, and let $f:S\to T$ and $g:T\to U$ be functions such that the function $g\circ f: S\to U$ is one-to-one (injective). Is it true that $f$ is onto or $g$ is onto?

Can anyone please help. I can't solve this problem.

Answer 1

In the following example $g\circ f$ is injective, but neither $f$ nor $g$ is surjective.

$$\{\bullet\}\xrightarrow{\;\;\;f\;\;\;}\begin{Bmatrix} \bullet\\ \star \end{Bmatrix}\xrightarrow{\;\;\;g\;\;\;}\begin{Bmatrix} \bullet\\ \star\\ \square \end{Bmatrix}$$