Problem
ALG-B3-M08-P008 Square as a Shift
#8
★★★☆☆ Level 3 of 5
Find all \(f(x)=ax+b\) such that \(f^2(x)=x+6\).
Compute \(a^2x+b(a+1)\).
We have \(f^2(x)=a^2x+b(a+1)\). Thus \(a^2=1\). If \(a=1\), then \(2b=6\), so \(b=3\). If \(a=-1\), the shift is \(0\), not \(6\). The answer is \(f(x)=x+3\).
Affine iteration with a shift.