16. Each of the following is not a function/mapping. In each case, demonstrate 24. In each of the following, decide if the given mapping is invertible. why. . If the mapping is invertible, exhibit an inverse mapping and verify that (a) f : R -+ R defined by f(x) = In(|x] -1) your mapping is the inverse. (b) a : Zg -+ Zg defined by . If the mapping is not invertible, prove that is isn't by demonstrating that if a is even the mapping is either not one-to-one or not onto. a (la]) = [ett ] if a is odd (a) f : R -+ R defined by f(x) = 2x + 5. 17. Define a mapping f : R - R by f(x) = 3x + 2. (b) g : Z x Z# + Q defined by g(m, n) = m for all (m, n) E Z x Z#. (Recall that Z# is the set of all nonzero integers.) (a) Prove that f is one-to-one. (c) h : M2(R) - M2(R) defined by h(A) = CA for each A E M2(RR) where (b) Prove that f is onto. 18. Define g : Z -+ Z by g(x) = 3x + 2. Prove that g is not onto. 19. Let B = {x ER|x 2 1} = [1, co). Define h : R -> B by h(x) = et. Prove that h maps R onto B. 20. Let a : A -+ B and B : B - C be mappings. (a) Prove that if a and B are 1 - 1 then Boa is 1 - 1. (b) Prove that if Bo a is onto then S is onto. 21. Solve each of the following equations for [x] in Zg. In each case, choose x so that 0