Problem
ALG-B3-M01-P012 Multiplying the argument
#12
★★★★☆ Level 4 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(xy)=x f(y)\) for all \(x,y\in\mathbb Q\).
Hint. Substitute \(y=1\).
With \(y=1\), \(f(x)=x f(1)\). Thus \(f(x)=cx\), where \(c\in\mathbb Q\). Check: \(f(xy)=cxy=x\cdot cy=x f(y)\).
Goal: separate guessing the answer from a complete proof and domain check.