Problem
ALG-B1-M08-P015 Iteration of a linear function
#15
★★★☆☆ Level 3 of 5
Find all linear functions \(f(x)=ax+b\) such that \(f(f(x))=4x+6\) for all \(x\), with the additional condition \(f(0)>0\).
Compute \(f(f(x))\) in terms of \(a,b\).
\(f(f(x))=a(ax+b)+b=a^2x+b(a+1)\). Hence \(a^2=4\) and \(b(a+1)=6\).
If \(a=2\), then \(3b=6\), so \(b=2\). If \(a=-2\), then \(-b=6\), so \(b=-6\), but then \(f(0)<0\). Only \(f(x)=2x+2\) works.
Tests iteration of a function as an algebraic object.