Question 28
Given a positive integer $n$ and the function $f: \mathbb{N} \to \mathbb{N}$ described by
$$
f(x) = \begin{cases} x/2 & , \text{if } x \text{ is even} \\ (x-1)/2 + 2^{n-1} & , \text{if } x \text{ is odd} \end{cases}.
$$
Determine the set $A = \{x \in \mathbb{N} \mid (\underbrace{f \circ f \circ \dots \circ f}_{n \text{ times } f})(x) = x\}$.
$$
f(x) = \begin{cases} x/2 & , \text{if } x \text{ is even} \\ (x-1)/2 + 2^{n-1} & , \text{if } x \text{ is odd} \end{cases}.
$$
Determine the set $A = \{x \in \mathbb{N} \mid (\underbrace{f \circ f \circ \dots \circ f}_{n \text{ times } f})(x) = x\}$.