Problem
ALG-B3-M02-P011 Iteration and invertibility
#11
★★★★★ Level 5 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(f(x)+y)=f(y)+x\) for all \(x,y\in\mathbb Q\).
Hint. First prove \(f(0)=0\), then \(f(f(x))=x\).
With \(x=0\), \(f(f(0)+y)=f(y)\). Thus shifting by \(f(0)\) does not change \(f\). With \(y=0\), \(f(f(x))=f(0)+x\). Comparing these relations gives \(f(0)=0\), hence \(f(f(x))=x\). Now replace \(x\) by \(f(t)\): \(f(t+y)=f(y)+f(t)\). Thus \(f\) is additive. On \(\mathbb Q\), \(f(q)=cq\), and \(f(f(q))=q\) gives \(c^2=1\). Answer: \(f(q)=q\) and \(f(q)=-q\).
Goal: choose the right first substitution and then verify the found function.