Problem
ALG-B3-M11-P012 Hidden Square Root of Three
#12
★★★★★ Level 5 of 5
Find all increasing \(f:\mathbb R\to\mathbb R\) such that \(f(x+f(y))=f(x)+3y\).
Prove injectivity and additivity.
If \(f(a)=f(b)\), comparing for \(y=a,b\) gives \(a=b\). With \(y=0\), \(f(0)=0\); with \(x=0\), \(f(f(y))=3y\). For \(t=f(y)\), we have \(f(t)=3y\), so \(f(x+t)=f(x)+f(t)\). An increasing additive function is linear: \(f(x)=cx\). Then \(c^2=3\), \(c>0\). The answer is \(f(x)=\sqrt{3}x\).
Combination of methods.