jmc

Question

If f(x) is defined for all integers x 0, f(1) = 1, and f(a + b) = f(a) + f(b) - 2f(ab)for all integers a, b 0, compute f(1986).

Accepted Answer

Setting b = 0 in the given functional equation, we get f(a) = f(a) + f(0) - 2f(0),so f(0) = 0. Setting b = 1 in the given functional equation, we get f(a + 1) = f(a) + f(1) - 2f(a) = f(1) - f(a).Then align* f(a + 2) &= f(1) - f(a + 1) &= f(1) - [f(1) - f(a)] &= f(a). align*Therefore, f(1986) = f(1984) = = f(2) = f(0) = 0.