Problem
ALG-B3-M08-P006 Additive Square
#6
★★★☆☆ Level 3 of 5
Find all additive \(f:\mathbb Q\to\mathbb Q\) such that \(f(f(x))=4x\).
On \(\mathbb Q\), an additive function has the form \(cx\).
Let \(f(x)=cx\). Then \(f(f(x))=c^2x\). We get \(c^2=4\), so \(c=2\) or \(c=-2\). The answer is \(f(x)=2x\), \(f(x)=-2x\).
Iteration becomes the square of the coefficient.