Question 43
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}$ satisfying
(i) $f(1) = 2008$,
(ii) $|f(x)| \le x^2 + 1004^2$, and
$$
(iii) \quad f\left(x + y + \frac{1}{x} + \frac{1}{y}\right) = f\left(x + \frac{1}{y}\right) + f\left(y + \frac{1}{x}\right)
$$
for all $x, y \gt 0$, where $\mathbb{R}$ is the set of all real numbers and $\mathbb{R}^+$ is the set of all positive real numbers.
(i) $f(1) = 2008$,
(ii) $|f(x)| \le x^2 + 1004^2$, and
$$
(iii) \quad f\left(x + y + \frac{1}{x} + \frac{1}{y}\right) = f\left(x + \frac{1}{y}\right) + f\left(y + \frac{1}{x}\right)
$$
for all $x, y \gt 0$, where $\mathbb{R}$ is the set of all real numbers and $\mathbb{R}^+$ is the set of all positive real numbers.