Problem 15.10. For each of the functions below, determine whether or not the func- tion is one-to-one and whether or not the function is onto. If the function is not one-to-one, give an explicit example to show what goes wrong. If it is not onto, determine the range. (a) Define f : R - R by f(x) = 1/(x2+1). (b) Define f : R - R by f(x) = sin(x). (Assume familiar facts about the sine function.) (c) Define f : Z x Z - Z by f(n, m) = nm. (d) Define f : R2 x R2 - R by f((x, y), (u, v)) = xu + yv. (Do you recognize this function?) (e) Define f : R2 x R2 - R by f((x, y), (u, v) ) = V(x - u)2+ (y- v)2. (Do you recognize this function?) (f) Let A and B be nonempty and b E B. Define f : A - A x B by f(a) = (a, b). (g) Let X be a nonempty set. Define f : "(X) - P(X) by f(A) = X \\ A. (h) Let B be a fixed proper subset of a nonempty set X. We define a function f : P (X) - (X) by f(A) = AnB. (i) Let f : R - R be defined by f (x ) = 2 - xifx