Let $f$: $A\rightarrow A$. Prove that if ($f$ ◦ $f$) ◦ $f$ is surjective, then $f$ is surjective.

Question

Let $f$: $A\rightarrow A$. Prove that if ($f$ ◦ $f$) ◦ $f$ is surjective, then $f$ is surjective.

Can someone help me out with this proof? Just had this question on an exam and trying to figure out if I got it right/what I did wrong. What I wrote on the test is below.

Since ($f$ ◦ $f$) ◦ $f$ is surjective, $\forall x\in A, \exists a$ such that $f(f(f(a))) = x$

Let $y = f(f(a)) \in A$

$\implies f(y) = f(f(f(a))) = x$

$\implies \forall x\in A, \exists y$ such that $f(y) = x$

$\therefore f$ is surjective

Answer 1

Yes, you got it right. Your proof is valid.