jmc

Question

Find the number of functions f(x), taking positive real numbers to real numbers, such that f(xy) = f(x) f(y) - 2for all positive real numbers x and y.

Accepted Answer

Setting y = 1, we get f(x) = f(1) f(x) - 2.Then f(1) f(x) - f(x) = 2, so f(x) = 2f(1) - 1.Thus, f(x) = c for some constant c for x 0. Then c = c^2 - 2, so c^2 - c - 2 = (c - 2)(c + 1) = 0. Thus, c = 2 or c = -1, giving us 2 functions f(x).