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Functional n-th root

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October 5, 2010
functionalgebra unsolvedalgebra

Problem Statement

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