Let X be the set of all positive such that `f(x+y) = f(xy)` for all `x ge 4, y ge 4`. If `f(8)= 9`, then f(9) is equal to.

Options

A. 8
B. 9
C. 81
D. 64

Explanation

To solve the problem, we need to find \( f(9) \) given the functional equation \( f(x+y) = f(xy) \) for all \( x \geq 4 \) and \( y \geq 4 \), and the information that \( f(8) = 9 \).

### Step-by-Step Solution:

1. **Express \( f(9) \)**:
We can express \( f(9) \) using the functional equation. We can write:
\[ f(9) = f(5 + 4) \]
Here, we set \( x = 5 \) and \( y = 4 \).

2. **Apply the functional equation**:
According to the functional equation:
\[ f(5 + 4) = f(5 \cdot 4) = f(20) \]
Thus, we have:
\[ f(9) = f(20) \]

3. **Express \( f(20) \)**:
Next, we can express \( f(20) \) again using the functional equation:
\[ f(20) = f(16 + 4) = f(16 \cdot 4) = f(64) \]

4. **Express \( f(64) \)**:
Now, we can express \( f(64) \):
\[ f(64) = f(8 + 8) = f(8 \cdot 8) = f(64) \]
This shows that \( f(64) = f(64) \), but we need to relate it back to \( f(8) \).

5. **Express \( f(64) \) in terms of \( f(8) \)**:
We can express \( f(64) \) as:
\[ f(64) = f(8 + 8) = f(8) + f(8) = 2f(8) \]
Since \( f(8) = 9 \), we have:
\[ f(64) = 2 \cdot 9 = 18 \]

6. **Relate \( f(20) \) back to \( f(8) \)**:
Now we can go back to \( f(20) \):
\[ f(20) = f(64) = 18 \]

7. **Finally, relate \( f(9) \) back to \( f(8) \)**:
Since \( f(9) = f(20) \), we conclude:
\[ f(9) = 18 \]

### Conclusion:
Thus, the value of \( f(9) \) is \( 18 \).