Problem
ALG-B3-M03-P006 Shift by one
#6
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)-f(0)\), \(f(0)=2\), and \(f(x+1)=f(x)+3\). Find \(f\).
Hint. First find \(f(1)\).
From \(x=0\) in the second condition, \(f(1)=5\). Now \(g=f-2\) is additive and \(g(1)=3\). Hence \(g(q)=3q\), so \(f(q)=3q+2\).
Goal: show which conditions actually force a function to be linear or affine.