Problem
ALG-B3-M10-P015 Reciprocal Argument with a Square
#15
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\) be additive and suppose that for all \(x\ne0\), \(f\left(\frac1x\right)=f(x)^2\). Find \(f\).
Again \(f(x)=cx\), then compare powers of \(x\).
Let \(f(x)=cx\). Then \(\frac c x=c^2x^2\) for all \(x\ne0\). If \(c=0\), we get the solution \(f=0\). If \(c\ne0\), then \(1=cx^3\) for all \(x\), impossible. The only answer is \(f\equiv0\).
Similar statement, completely different result.