Functions

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions f: ℕ → ℤ and g: ℤ → ℕ defined by

$$ f(n) = \begin{cases} \frac{(n + 1)}{2} & \text{if } n \text{ is odd,} \\ \frac{(4-n)}{2} & \text{if } n \text{ is even,} \end{cases} $$

and

$$ g(n) = \begin{cases} 3 + 2n & \text{if } n \ge 0 , \\ -2n & \text{if } n < 0 . \end{cases} $$

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?