A function f(x) satisfies f(x) + f(1/(1-x)) = x for all x ≠ 0, 1. Find f(2).
Correct: D
We are given f(x) + f(1/(1-x)) = x. Let x = 2, then f(2) + f(1/(1-2)) = f(2) + f(-1) = 2. Now, let x = -1, then f(-1) + f(1/(1-(-1))) = f(-1) + f(1/2) = -1. Next, let x = 1/2, then f(1/2) + f(1/(1-1/2)) = f(1/2) + f(2) = 1/2. We have the following system of equations: 1) f(2) + f(-1) = 2; 2) f(-1) + f(1/2) = -1; 3) f(1/2) + f(2) = 1/2. Subtract equation (2) from (1) to get: f(2) - f(1/2) = 3. Adding this to equation (3), we have: 2f(2) = 3 + 1/2 = 7/2. Therefore, f(2) = 7/4.