Problem
ALG-B3-M05-P004 Shifted Additivity
#4
★★☆☆☆ Level 2 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)=f(x)+f(y)+6\) for all \(x,y\).
Try \(g(x)=f(x)+6\).
Let \(g(x)=f(x)+6\). Then \(g(x+y)=f(x+y)+6=f(x)+f(y)+12=g(x)+g(y)\). Hence \(g(x)=ax\), where \(a\in\mathbb Q\). Therefore \(f(x)=ax-6\). All such functions work.
The shift is chosen so that the constant disappears.