Problem
ALG-B3-M05-P013 Additive Iteration
#13
★★★★☆ Level 4 of 5
Find all additive \(f:\mathbb Q\to\mathbb Q\) such that \(f(f(x))=4x\).
Additivity on \(\mathbb Q\) gives \(f(x)=cx\).
Since \(f\) is additive on \(\mathbb Q\), \(f(x)=cx\), where \(c\in\mathbb Q\). Then \(f(f(x))=c^2x\). The condition gives \(c^2=4\), hence \(c=2\) or \(c=-2\). The answers are \(f(x)=2x\) and \(f(x)=-2x\).
Composition after Cauchy often becomes an equation for the coefficient.