Problem
ALG-B3-M11-P011 Golden System
#11
★★★★★ Level 5 of 5
Let \(f\) be continuous, additive, and satisfy \(f(x)f(y)=f(xy)+xy\). Find \(f\).
Linearity, then a quadratic equation.
Let \(f(x)=cx\). Then \(c^2xy=cxy+xy\), so \(c^2=c+1\). The answer is \(f(x)=\frac{1+\sqrt{5}}{2}x\) or \(f(x)=\frac{1-\sqrt{5}}{2}x\).
System with an irrational coefficient.