Proving if this function is bijective

Question

Suppose we have a function: $f:A\rightarrow A$ such that $f(f(a)) = a$ for all $a$. Prove that this is bijective.

For me to prove this is one to one: $f(f(b))=f(f(a)) \rightarrow f^-1f(f(b))=f^-1f(f(a)) \rightarrow f(b) = f(a)$, thus one to one has been proven.

Confused how to do onto...

Answer 1

$f$ is onto because $a = f(f(a)) \in f(A)$