Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 1E / i

This can fail. Let $X=\{1\}$ and $Y=\{a,b\}$, put $f(1)=a$, and let $g(a)=g(b)=1$. Then the function composition $g\circ f$ is the identity function on $X$, hence is a surjective function, although $f$ is not surjective.