Let ℕ denote the set of all natural numbers, and ℤ denote the set...

Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ and $g: \mathbb{Z} \rightarrow \mathbb{N}$ 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 $g(n)=\left\{\begin{array}{cc}3+2 n & \text{\text{ if } } n \geq 0 \\ -2 n & \text{\text{ if } } n <0\end{array}$\right. 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?