Problem
ALG-B3-M03-P004 Coefficients in the argument
#4
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(2x+3y)=2f(x)+3f(y)\), \(f(1)=7\). Find \(f\).
Hint. First prove additivity by choosing \(x=u/2\), \(y=v/3\).
From \(x=y=0\), \(f(0)=0\). For any \(u,v\in\mathbb Q\), take \(x=u/2\), \(y=v/3\): then \(f(u+v)=f(u)+f(v)\). Hence \(f(q)=7q\).
Goal: show which conditions actually force a function to be linear or affine.