 Solution

One example is the identity function, $f(n)=n$. A second is the permutationBoth are bijections of $\mathbb{N}$, whichever convention includes or excludes zero. For the second, applying $f$ three times returns every input to itself. Thus $f^3=\mathrm{id}$, and composing with the inverse function gives $f^2=f^{-1}$. This permutation cube identity criterion also shows why three-cycles work whereas a nontrivial two-cycle would not.