CAT 2008 | Question: 13

Question

Let $f(x) $ be a function satisfying $f(x) f(y) =f(xy)$ for all real $x, y$. If $f(2)=4$, then what is the value of $f\left(\frac{1}{2}\right)$

• A. $0$

• B. $\frac{1}{4}$

• C. $\frac{1}{2}$

• D. $1$

• E. cannot be determined

Answer 1

Option $B$ is the correct answer.

 given, $f\left ( 2 \right ) = 4 \\$.
$f\left ( 4 \right ) = f\left ( 2 \times 2 \right ) \\$

but since $f\left ( x \cdot y \right ) = f\left ( x \right ) \times f\left ( y \right ) \\$
$\Rightarrow f\left ( 4 \right ) = f\left ( 2 \right ) \times f\left ( 2 \right ) \\$
$\Rightarrow f\left ( 4 \right ) = 16 \left (=4 \times 4 \right ).\\$
$f\left ( 2 \right ) = f\left ( 4 \times \frac{1}{2} \right )\\$
$\Rightarrow f\left ( 2 \right ) = f\left ( 4   \right )\times f\left ( \frac{1}{2} \right )\\$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{f\left ( 2 \right )}{f\left ( 4   \right )}\\$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{4}{16}$
$\Rightarrow f\left ( \frac{1}{2} \right ) = \frac{1}{4}$