Problem
ALG-B3-M10-P014 Additivity and Reciprocal Argument
#14
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\) be additive and suppose that for all \(x\ne0\), \(f\left(\frac1x\right)=\frac{f(x)}{x^2}\). Find \(f\).
Let \(f(x)=cx\).
Additivity on \(\mathbb Q\) gives \(f(x)=cx\). The left-hand side is \(\frac c x\), and the right-hand side is \(\frac{cx}{x^2}=\frac c x\). The condition does not restrict \(c\). The answer is all \(f(x)=cx\), \(c\in\mathbb Q\).
Sometimes the second condition is compatible with the whole family.