Let R denote the set of real numbers. Let f : R - R - R be a bijective function defined by g(C) = {f(x) | x in C} for ...

Question

Let R denote the set of real numbers. Let f : R -> R -> R be a bijective function defined by g(C) = {f(x) | x in C} for all subsets C of R. The inverse function of f is given by

A. f^(-1)(x, y) = (x - y, x + y)
B. f^(-1)(x, y) = (x - y, x + y) / 2
C. f^(-1)(x, y) = ((x + y)/2, (x - y)/2)
D. f^(-1)(x, y) = (2(x - y), 2(x + y))

Accepted Answer

f^(-1)(x, y) = ((x+y)/2, (x-y)/2) The function f maps a pair (a, b) to (a+b, a-b). To find the inverse, we set (x, y) = (a+b, a-b) and solve for a and b in terms of x and y. Adding the two equations gives x + y = 2a, so a = (x+y)/2; subtracting gives x - y = 2b, so b = (x-y)/2. This gives f^(-1)(x,y) = ((x+y)/2, (x-y)/2).