Problem
ALG-B1-M08-P019 Additivity and square
#19
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\) be additive and satisfy \(f(x^2)=f(x)^2\) for all \(x\in\mathbb Q\). Find all such functions.
First use additivity on \(\mathbb Q\): \(f(x)=cx\).
By additivity on \(\mathbb Q\), \(f(x)=cx\), where \(c=f(1)\). The condition gives \(c x^2=c^2 x^2\) for all \(x\). At \(x=1\), \(c=c^2\), so \(c=0\) or \(c=1\).
Both functions work: \(f(x)=0\) and \(f(x)=x\).
A good problem on adding a second condition to Cauchy.