Problem
ALG-B3-M10-P020 Full Parametric Check
#20
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)=f(x)+f(y)+a xy\), and \(f(f(x))=x+a\).
If \(a e0\), compare the degree of the composition. If \(a=0\), solve the linear case.
From the first condition, \(f(x)=\frac a2x^2+cx\). If \(a\ne0\), then \(f\) is quadratic and \(f(f(x))\) has degree \(4\), while \(x+a\) has degree \(1\). Hence \(a=0\). Then \(f(x)=cx\), and the condition becomes \(c^2x=x\). Therefore \(c^2=1\). The answer is \(a=0\), \(f(x)=x\) or \(f(x)=-x\).
Final task of the module: structure, parameter, composition, verification.