Problem
ALG-B3-M01-P009 Shift with a given value
#9
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+1\), and \(f(1)=0\). Find \(f\).
Hint. Put \(g(x)=f(x)+1\).
Then \(g\) is additive and \(g(1)=1\). On \(\mathbb Q\), \(g(q)=q\). Hence \(f(q)=q-1\). Check: \(x+y-1=(x-1)+(y-1)+1\).
Goal: separate guessing the answer from a complete proof and domain check.