Option $B$ is the correct answer.
given, $f\left ( 2 \right ) = 4 \\$.
$f\left ( 4 \right ) = f\left ( 2 \times 2 \right ) \\$
but since $f\left ( x \cdot y \right ) = f\left ( x \right ) \times f\left ( y \right ) \\$
$\Rightarrow f\left ( 4 \right ) = f\left ( 2 \right ) \times f\left ( 2 \right ) \\$
$\Rightarrow f\left ( 4 \right ) = 16 \left (=4 \times 4 \right ).\\$
$f\left ( 2 \right ) = f\left ( 4 \times \frac{1}{2} \right )\\$
$\Rightarrow f\left ( 2 \right ) = f\left ( 4 \right )\times f\left ( \frac{1}{2} \right )\\$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{f\left ( 2 \right )}{f\left ( 4 \right )}\\$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{4}{16}$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{1}{4}$