Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 4 / 7D / a / Solution

Induction shows that every iterate $f^n$ is injective. Applying $f$ repeatedly givesIf $f^k(X)=f^{k+1}(X)$, applying $f$ inductively gives equality with every later image. Set $A=f^k(X)$. Then $f(A)=A$, so the restriction $f:A\to A$ is surjective, and it remains injective; hence it is bijective.