Problem
ALG-B3-M01-P005 Additivity on rationals
#5
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\), and \(f(3)=12\). Find \(f(q)\) for all \(q\in\mathbb Q\).
Hint. First find \(f(1)\).
Additivity gives \(f(3)=3f(1)\), so \(f(1)=4\). For integers \(n\), \(f(n)=4n\), and from \(m f(m/n)=f(m)=4m\), \(f(m/n)=4m/n\). Answer: \(f(q)=4q\).
Goal: separate guessing the answer from a complete proof and domain check.