MathDB

Let

n, a, b

be integers with

n \geq 2

and

a \notin \{0, 1\}

and let

u(x) = ax + b

be the function defined on integers. Show that there are infinitely many functions

f : \mathbb{Z} \rightarrow \mathbb{Z}

such that

f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)

for all

x

. If

a = 1

, show that there is a

b

for which there is no

f

with

f_n(x) \equiv u(x)

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