First, we prove the function g is an injection (i.e., g is one-to-one). We need to prove that if g(m) = g(n) --(1), where m and n are two integers, then m = n --(2). Since g(m) (and also g(n)) can be defined by two different formulas depending whether the argument m (and n) is even or odd, we consider the following 3 cases: (Case 1) Both m and n are even. In this case, (1) implies 1 – m = 1 – n...