Question

Let $f : \bullet \to \bullet$ satisfy $f (x) f ( y) = f (xy)$ for all real numbers $x$ and $y$. If $f (2)= 4$, then $f \left( \frac{1}{2} \right) =$

• A. 0
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. 1

Answer 1

Answer: (B)

$\frac{1}{4}$

Answer 2

Given, $f: \bullet \rightarrow \bullet$ $f(x) f(y)=f(x y)$ ...(i) On taking $x=1,\, y=1$ $f()1 f( 1 )=f( 1 \cdot 1 )=f( 1 )^{2}=f( 1 )=f( 1 )= 1$ Now, $x=2,\, y=\frac{1}{2}$, then from E (i) $f(2) f\left(\frac{1}{2}\right)=f\left(2 \cdot \frac{1}{2}\right)$ $\Rightarrow 4 \cdot f\left(\frac{1}{2}\right)=f(1) [\because f(2)=4]$ $\Rightarrow f\left(\frac{1}{2}\right)=\frac{1}{4} f(1)$ On putting the value of $f(1)$, $\Rightarrow f\left(\frac{1}{2}\right)=\frac{1}{4} \cdot 1 =\frac{ 1 }{4}$