Problem
ALG-B3-M03-P007 Midpoint equality
#7
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}2\), \(f(0)=1\), \(f(1)=4\). Find \(f\).
Hint. Consider \(g(x)=f(x)-1\).
For \(g=f-1\), \(g((x+y)/2)=(g(x)+g(y))/2\), \(g(0)=0\), \(g(1)=3\). On rationals, the midpoint relation gives additivity: \(g(u+v)=g(u)+g(v)\) after taking \(x=2u\), \(y=2v\) and using \(g(2t)=2g(t)\). Hence \(g(q)=3q\), and \(f(q)=3q+1\).
Goal: show which conditions actually force a function to be linear or affine.