ILL 1983 Problem 20

Let $f$ and $g$ be functions from the set $A$ to the same set $A$ . We define $f$ to be a functional $n$ -th root of $g$ ( $n$ is a positive integer) if $f^n(x) = g(x)$ , where $f^n(x) = f^{n-1}(f(x)).$ (a) Prove that the function $g : \mathbb R \to \mathbb R, g(x) = 1/x$ has an infinite number of $n$ -th functional roots for each positive integer $n.$ (b) Prove that there is a bijection from $\mathbb R$ onto $\mathbb R$ that has no nth functional root for each positive integer $n.$