Problem
ALG-B3-M03-P015 Composition with two shifts
#15
★★★★★ Level 5 of 5
Find all affine \(f(x)=ax+b\) such that \(f(f(x)+1)=9x+5\) for all \(x\).
Hint. Expand \(f(ax+b+1)\).
We have \(f(f(x)+1)=a(ax+b+1)+b=a^2x+ab+a+b\). Thus \(a^2=9\) and \(b(a+1)+a=5\). If \(a=3\), then \(4b+3=5\), so \(b=1/2\). If \(a=-3\), then \(-2b-3=5\), so \(b=-4\). Both work.
Goal: show which conditions actually force a function to be linear or affine.