Problem
ALG-B3-M07-P008 Compositional Involution
#8
★★★☆☆ Level 3 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=x\).
Compare degrees.
If \(P\) is constant, impossible. Let \(\deg P=d\). Then \(\deg P(P(x))=d^2\), while the right-hand side has degree \(1\). Thus \(d=1\). Let \(P(x)=ax+b\). Then \(P(P(x))=a^2x+b(a+1)\). We get \(a^2=1\), \(b(a+1)=0\). The answer is \(P(x)=x\) and \(P(x)=-x+b\), where \(b\in\mathbb R\).
Composition reduces to the linear case.