Problem
ALG-B1-M10-P014 Additivity
#14
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)\), \(f(3)=12\). Find \(f\left(\frac{5}{2}\right)\).
First find \(f(1)\).
\(f(3)=3f(1)=12\), so \(f(1)=4\). Then \(f(q)=4q\) for rational \(q\), hence \(f\left(\frac{5}{2}\right)=10\).
In the mixed module, the functional equation should be recognized quickly.