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((1)/(2))`?

Options

A. 0
B. `(1)/(4)`
C. `(1)/(2)`
D. cannot be determined

Answer

To solve the problem, we need to find the value of \( f\left(\frac{1}{2}\right) \) given that the function \( f(x) \) satisfies the property \( f(x) f(y) = f(xy) \) for all real \( x \) and \( y \), and that \( f(2) = 4 \).

### Step-by-Step Solution:

1. **Understand the Functional Equation**:
The equation \( f(x) f(y) = f(xy) \) suggests that the function \( f \) has a multiplicative property.

2. **Substituting \( y = 1 \)**:
Let's substitute \( y = 1 \) into the functional equation:
\[ f(x) f(1) = f(x \cdot 1) = f(x) \]
This implies:
\[ f(x) f(1) = f(x) \]
If \( f(x) \neq 0 \), we can divide both sides by \( f(x) \):
\[ f(1) = 1 \]

3. **Substituting \( x = 2 \) and \( y = \frac{1}{2} \)**:
Now, we want to find \( f\left(\frac{1}{2}\right) \). We substitute \( x = 2 \) and \( y = \frac{1}{2} \) into the functional equation:
\[ f(2) f\left(\frac{1}{2}\right) = f\left(2 \cdot \frac{1}{2}\right) = f(1) \]
We know \( f(2) = 4 \) and \( f(1) = 1 \):
\[ 4 \cdot f\left(\frac{1}{2}\right) = 1 \]

4. **Solving for \( f\left(\frac{1}{2}\right) \)**:
Now we can solve for \( f\left(\frac{1}{2}\right) \):
\[ f\left(\frac{1}{2}\right) = \frac{1}{4} \]

### Final Answer:
Thus, the value of \( f\left(\frac{1}{2}\right) \) is \( \frac{1}{4} \).