Question 27
Let $\mathbb{Q}^{+}$ denote the set of positive rational numbers. Show that there exists one and only one function $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ satisfying the following conditions:
(i) If $0 \lt q \lt \frac{1}{2}$ then $f(q) = 1 + f\left(\frac{q}{1 - 2q}\right)$.
(ii) If $1 \lt q \leq 2$ then $f(q) = 1 + f(q - 1)$.
(iii) $f(q) \cdot f\left(\frac{1}{q}\right) = 1$ for all $q \in \mathbb{Q}^{+}$.
(i) If $0 \lt q \lt \frac{1}{2}$ then $f(q) = 1 + f\left(\frac{q}{1 - 2q}\right)$.
(ii) If $1 \lt q \leq 2$ then $f(q) = 1 + f(q - 1)$.
(iii) $f(q) \cdot f\left(\frac{1}{q}\right) = 1$ for all $q \in \mathbb{Q}^{+}$.