jmc

Question

Let f : R R be a function such that f(1) = 1 and f(x + f(y + z)) + f(f(x + y) + z) = 2yfor all real numbers x, y, and z. Let n be the number of possible values of f(5), and let s be the sum of all possible values of f(5). Find n s.

Accepted Answer

Setting x = z = 0, we get 2f(f(y)) = 2y,so f(f(y)) = y for all y. Setting y = z = 0, we get f(x + f(0)) + f(f(x)) = 0.Since f(f(x)) = x, f(x + f(0)) + x = 0,so f(x + f(0)) = -x. Let w = x + f(0), so f(w) = f(0) - w.Since x can represent any number, this holds for all w. Hence, f(x) = c - x for some constant c. And since f(1) = 1, we must have f(x) = 2 - x. We can check that this function works. Thus, n = 1 and s = 2 - 5 = -3, so n s = -3.