Problem
ALG-B3-M01-P002 Shift to an additive function
#2
★★☆☆☆ Level 2 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)=f(x)+f(y)-2\) for all \(x,y\in\mathbb Q\).
Hint. Consider \(g(x)=f(x)-2\).
Then \(g(x+y)=g(x)+g(y)\). On \(\mathbb Q\), such a function has the form \(g(q)=cq\), where \(c=g(1)\in\mathbb Q\). Hence \(f(q)=cq+2\). Verification is direct.
Goal: separate guessing the answer from a complete proof and domain check.