Problem
ALG-B3-M02-P001 Linear shift
#1
★★☆☆☆ Level 2 of 5
Find all \(f:\mathbb R\to\mathbb R\) such that \(f(x+y)=f(x)+2y\) for all \(x,y\in\mathbb R\).
Hint. Substitute \(x=0\).
With \(x=0\), \(f(y)=f(0)+2y\). Hence \(f(x)=2x+c\). Check: \(f(x+y)=2x+2y+c=f(x)+2y\). Answer: \(f(x)=2x+c\), where \(c\in\mathbb R\).
Goal: choose the right first substitution and then verify the found function.