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) = (n + 1)/2 if n is odd and f(n) = (4 − n)/2 if n is even, and g(n) = 3 + 2n if n ≥ 0 and g(n) = −2n if n < 0. Define (g ∘ f)(n) = g(f(n)) for all n ∈ ℕ, and (f ∘ g)(n) = f(g(n)) for all n ∈ ℤ. Then which of the following statements is (are) TRUE?