Answer
$\frac{1+2\sqrt 6}{6}$
Work Step by Step
Step 1. Using the given figures, we have $x=\sqrt {4-1}=\sqrt 3$ and $y=\sqrt {1-1/9}=-\frac{2\sqrt 2}{3}$
Step 2. We have $sin(\alpha)=\frac{1}{2}, cos(\alpha)=\frac{\sqrt 3}{2}, sin(\beta)=-\frac{2\sqrt 2}{3}, cos(\beta)=\frac{1}{3}$
Step 3. $f(\alpha-\beta)=sin(\alpha-\beta)=sin(\alpha)cos(\beta)-cos(\alpha)sin(\beta)=(\frac{1}{2})(\frac{1}{3})-(\frac{\sqrt 3}{2})(-\frac{2\sqrt 2}{3})=\frac{1+2\sqrt 6}{6}$