jmc

Question

The function f(x) takes positive real numbers to real numbers, such that xf(y) - yf(x) = f ( xy )for all positive real numbers x and y. Find all possible values of f(100). Enter all possible values, separated by commas.

Accepted Answer

Setting x = y, we get 0 = f(1).Setting y = 1, we get xf(1) - f(x) = f(x),so 2f(x) = 0, which means f(x) = 0 for all x. (Note that this function satisfies the given functional equation.)