Problem
ALG-B3-M03-P016 Affineness from two rules
#16
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)-f(0)\), and \(f(3x)=3f(x)-4\). Find \(f\).
Hint. Write \(f=g+c\), where \(g\) is additive.
Let \(c=f(0)\), \(g=f-c\). Then \(g\) is additive, and \(f(3x)=g(3x)+c=3g(x)+c\). The right side of the second condition is \(3(g(x)+c)-4=3g(x)+3c-4\). Hence \(c=3c-4\), so \(c=2\). The additive part on \(\mathbb Q\) is \(g(q)=kq\). Answer: \(f(q)=kq+2\), \(k\in\mathbb Q\).
Goal: show which conditions actually force a function to be linear or affine.