Problem
ALG-B3-M10-P012 Parameter and Involution
#12
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)=f(x)+f(y)+a xy\), and \(f(f(x))=x\).
If \(a e0\), the function is quadratic.
From the first condition, \(f(x)=\frac a2x^2+cx\). If \(a\ne0\), then \(f\) has degree \(2\), and \(f(f(x))\) has degree \(4\), impossible. Hence \(a=0\). Then \(f(x)=cx\), and \(c^2=1\). The answer is \(a=0\), \(f(x)=x\) or \(f(x)=-x\).
The parameter is eliminated by composition.