Problem
ALG-B3-M02-P013 Two halves
#13
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)+f(x-y)=2f(x)+8y^2\), \(f(0)=0\), \(f(1)=2\). Find \(f\).
Hint. Subtract \(2x^2\).
Put \(g(x)=f(x)-2x^2\). Then \(g(x+y)+g(x-y)=2g(x)\). With \(x=0\), \(g(y)+g(-y)=0\). With \(x=(u+v)/2\), \(y=(u-v)/2\), \(g(u)+g(v)=2g((u+v)/2)\). On \(\mathbb Q\), Jensen's equation with \(g(0)=0\), \(g(1)=0\) gives \(g\equiv0\). Hence \(f(q)=2q^2\).
Goal: choose the right first substitution and then verify the found function.