Problem
ALG-B1-M08-P016 A rational value
#16
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\), \(f(2)=5\). Find \(f\left(\frac{7}{3}\right)\).
First find \(f(1)\), then \(f\left(\frac{1}{3}\right)\).
Since \(f(2)=2f(1)=5\), we have \(f(1)=\frac{5}{2}\). Next, \(3f\left(\frac{1}{3}\right)=f(1)=\frac{5}{2}\), so \(f\left(\frac{1}{3}\right)=\frac{5}{6}\). Therefore \(f\left(\frac{7}{3}\right)=7\cdot\frac{5}{6}=\frac{35}{6}\).
A small extension of the example on \(\mathbb Q\).