Let R denote the set of real numbers. Let f : R x R - R x R be a bijective function defined by f(x, y) = (x + y, x - y)...

Question

Let R denote the set of real numbers. Let f : R x R -> R x R be a bijective function defined by f(x, y) = (x + y, x - y). The inverse function of f is given by

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

Accepted Answer

C. f^{-1}(x, y) = ((x + y)/2, (x - y)/2) To find the inverse of a bijective function f(x, y) = (u, v) where u = x + y and v = x - y, we solve the system for x and y in terms of u and v. Adding the two equations gives 2x = u + v, so x = (u + v)/2. Subtracting gives 2y = u - v, so y = (u - v)/2. Thus f^{-1}(u, v) = ((u + v)/2, (u - v)/2).