jmc

Setting $x=1$ and $y=1,$ we get $$f(2)=f(1)f(1),$$so $f(1{)}^2=9.$ Then $f(1)=\pm 3.$

Setting $x=\frac{1}{2}$ and $y=\frac{1}{2},$ we get $$f(1)=f\left(\frac{1}{2}\right)f\left(\frac{1}{2}\right)=f{\left(\frac{1}{2}\right)}^2\ge 0,$$so $f(1)=3.$

Setting $x=1$ and $y=2,$ we get $$f(3)=f(1)f(2)=27.$$Setting $x=2$ and $y=3,$ we get $$f(5)=f(2)f(3)=\boxed{243}.$$