Problem
ALG-B3-M01-P018 Squares force linearity
#18
★★★★★ Level 5 of 5
Let \(f:\mathbb R\to\mathbb R\) be additive and satisfy \(f(x^2)=x f(x)\) for all \(x\in\mathbb R\). Find all such functions.
Hint. Apply the condition to \((x+y)^2\).
On one hand, \(f((x+y)^2)=(x+y)(f(x)+f(y))\). On the other hand, by additivity, \(f(x^2+2xy+y^2)=xf(x)+2f(xy)+yf(y)\). Hence \(2f(xy)=xf(y)+yf(x)\). With \(y=1\), \(2f(x)=xf(1)+f(x)\), so \(f(x)=cx\), where \(c=f(1)\). Every function \(f(x)=cx\) works.
Goal: separate guessing the answer from a complete proof and domain check.