Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$(a) Find $f(1)$. (b) Find $f(x)$.
Source: 2015 Cuba 2.1
Tags: algebra, functional, functional equation
Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$(a) Find $f(1)$. (b) Find $f(x)$.