Problem
ALG-B3-M08-P009 Cube as a Shift
#9
★★★☆☆ Level 3 of 5
Find all \(f(x)=ax+b\) such that \(f^3(x)=x+6\).
For real \(a\), from \(a^3=1\) we get \(a=1\).
The coefficient of \(x\) in \(f^3(x)\) is \(a^3\), so \(a^3=1\), hence \(a=1\). Then \(f(x)=x+b\), and \(f^3(x)=x+3b\). We get \(3b=6\), so \(b=2\). The answer is \(f(x)=x+2\).
Useful task on third iteration.