Problem
ALG-B3-M04-P008 A Coefficient from the Image
#8
★★★☆☆ Level 3 of 5
Find all surjective \(f:\mathbb R\to\mathbb R\) such that \(f(x+f(y))=f(x)+2f(y)\) for all \(x,y\).
Make \(t=f(y)\) arbitrary.
By surjectivity, \(t=f(y)\) is arbitrary. Hence \(f(x+t)=f(x)+2t\) for all \(x,t\). Taking \(x=0\), we get \(f(t)=f(0)+2t\). Therefore \(f(x)=2x+c\). Checking: \(f(x+f(y))=2x+2f(y)+c=f(x)+2f(y)\). All functions \(f(x)=2x+c\) work.
Shows that not every module example reduces to \(x+c\); the coefficient may also come from the condition.