Problem
ALG-B3-M05-P006 Additive and Multiplicative
#6
★★★☆☆ Level 3 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)=f(x)+f(y)\) and \(f(xy)=f(x)f(y)\).
First use additivity on \(\mathbb Q\).
Additivity gives \(f(x)=cx\). Then \(f(xy)=cxy\), while \(f(x)f(y)=c^2xy\). Hence \(cxy=c^2xy\) for all \(x,y\), so \(c=0\) or \(c=1\). The answers are \(f(x)=0\) and \(f(x)=x\).
The zero function is a genuine solution.