Problem
ALG-B3-M11-P019 Two Functions
#19
★★★★★ Level 5 of 5
Let \(f,g:\mathbb Q\to\mathbb Q\) be additive, \(f(g(x))=x\), and \(f(2)=6\). Find \(f,g\).
Let \(f(x)=ax\), \(g(x)=bx\).
From \(f(2)=6\), \(2a=6\), so \(a=3\). The condition \(f(g(x))=x\) gives \(ab=1\), hence \(b=\frac13\). The answer is \(f(x)=3x\), \(g(x)=\frac{x}{3}\).
System of functions in a mixed set.