Problem
ALG-B3-M08-P007 Additive Cube
#7
★★★☆☆ Level 3 of 5
Find all additive \(f:\mathbb Q\to\mathbb Q\) such that \(f^3(x)=8x\).
If \(f(x)=cx\), then \(f^3(x)=c^3x\).
Additivity on \(\mathbb Q\) gives \(f(x)=cx\). The condition \(f^3(x)=8x\) gives \(c^3=8\), hence \(c=2\). The answer is \(f(x)=2x\).
Cubic iteration on rationals.