MathDB

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.

20