Question

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

• A. \( 0 \)
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. \( 1 \)
• E. \( 2 \)

Hint:

Always evaluate functional equations at special values like \( x=1 \).

Answer 1

The Correct Option is B

Concept: Given functional equation: \[ f(xy) = f(x)f(y) \]

Step 1: Put \( x=2, y=\frac{1}{2} \).
\[ f(1) = f(2)f\left(\frac{1}{2}\right) \]

Step 2: Find \( f(1) \).
\[ f(1) = f(1 \cdot 1) = f(1)^2 \Rightarrow f(1)=1 \]

Step 3: Substitute.
\[ 1 = 4 \cdot f\left(\frac{1}{2}\right) \Rightarrow f\left(\frac{1}{2}\right) = \frac{1}{4} \]