Problem
ALG-B3-M03-P018 Iteration with no solutions
#18
★★★★★ Level 5 of 5
Prove that there is no affine function \(f(x)=ax+b\) such that \(f(f(x))=x+1\) and \(f(0)=0\).
Hint. The condition \(f(0)=0\) immediately gives \(b=0\).
From \(f(0)=0\), \(b=0\), so \(f(x)=ax\). Then \(f(f(x))=a^2x\), which cannot equal \(x+1\) for all \(x\), because the right side has constant term \(1\). No solutions.
Goal: show which conditions actually force a function to be linear or affine.