Question 161
Let $k$ be a natural number. For every function $f: \mathbb{N} \rightarrow \mathbb{N}$, let the sequence of functions $\left(f_{m}\right)_{m \geqslant 1}$ be defined by $f_{1}=f$ and $f_{m+1}=f \circ f_{m}$ for $m \geqslant 1$. The function $f$ is $k$-fine if for all $n \in \mathbb{N}$ we have
$$
f_{k}(n)=f(n)^{k}
$$
a) For which $k$ does there exist a $1$-$1$ (injective) $k$-fine function $f$?
b) For which $k$ does there exist an onto (surjective) $k$-fine function $f$?
$$
f_{k}(n)=f(n)^{k}
$$
a) For which $k$ does there exist a $1$-$1$ (injective) $k$-fine function $f$?
b) For which $k$ does there exist an onto (surjective) $k$-fine function $f$?