Problem
ALG-B3-M05-P003 A Rational Line
#3
★★☆☆☆ Level 2 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)=f(x)+f(y)\) and \(f(3)=12\).
On \(\mathbb Q\), an additive function is determined by \(f(1)\).
By additivity on \(\mathbb Q\), \(f(q)=qf(1)\). From \(f(3)=3f(1)=12\), we get \(f(1)=4\). The answer is \(f(q)=4q\).
A short task reinforcing the fact over \(\mathbb Q\).