Problem
ALG-B3-M03-P008 Iteration of an affine function
#8
★★★★☆ Level 4 of 5
Find all affine functions \(f(x)=ax+b\) such that \(f(f(x))=4x+3\) for all \(x\).
Hint. Compare the coefficient of \(x\) and the constant term.
We have \(f(f(x))=a(ax+b)+b=a^2x+b(a+1)\). Thus \(a^2=4\), \(b(a+1)=3\). If \(a=2\), then \(b=1\); if \(a=-2\), then \(b=-3\). Answer: \(2x+1\) and \(-2x-3\).
Goal: show which conditions actually force a function to be linear or affine.