Question 6

Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f: \mathbb{N} \to \mathbb{Z}$ and $g: \mathbb{Z} \to \mathbb{N}$ defined by $f(n) = \begin{cases} (n + 1)/2 & \text{if } n \text{ is odd} \\ (4 - n)/2 & \text{if } n \text{ is even} \end{cases}$ and $g(n) = \begin{cases} 3 + 2n & \text{if } n \geq 0 \\ -2n & \text{if } n < 0 \end{cases}$ Define $(g \circ f)(n) = g(f(n))$ for all $n \in \mathbb{N}$, and $(f \circ g)(n) = f(g(n))$ for all $n \in \mathbb{Z}$. Then which of the following statements is (are) TRUE?