jmc

Question

Let f(x) be a function such that f(0) = 1 and f(xy) = f ( x^2 + y^22 ) + (x - y)^2for all real numbers x and y. Find f(x).

Accepted Answer

Setting y = 0, we get f(0) = f ( x^22 ) + x^2.Hence, f(u) = 1 - 2u for all u 0. Setting y = 1, we get f(x) = f ( x^2 + 12 ) + (x - 1)^2 = 1 - 2 x^2 + 12 + (x - 1)^2 = 1 - 2x.