Problem
ALG-B3-M10-P010 Two Inverse Additive Functions
#10
★★★★★ Level 5 of 5
Let \(f,g:\mathbb Q\to\mathbb Q\) be additive and satisfy \(f(g(x))=x\), \(g(f(x))=x\). Find all pairs.
Let \(f(x)=ax\), \(g(x)=bx\).
On \(\mathbb Q\), \(f(x)=ax\), \(g(x)=bx\). The conditions give \(ab=1\). Hence \(a\in\mathbb Q\setminus\{0\}\), \(b=\frac1a\). The answer is \(f(x)=ax\), \(g(x)=\frac{x}{a}\).
System of two functions.