jmc

Question

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

Accepted Answer

Setting x = y, we get f(0) = 0. Setting x = -1 and y = 0, we get f(1) = -f(-1),so f(-1) = -1. Setting y = 1 and y = -1, we get align* f(x^2 - 1) &= (x - 1) (f(x) + 1), f(x^2 - 1) &= (x + 1) (f(x) - 1), align*respectively. Hence, (x - 1) (f(x) + 1) = (x + 1) (f(x) - 1), which simplifies to f(x) = x. We can check that this function works. Therefore, n = 1 and s = 2, so n s = 2.