The domain of `f(x)` such that the `f(x) = [[x + 1/2]] / [[x - 1/2]]` is prime is `[X_1, X_2)`, then the value of `2(x_1 ^2 + x_2 ^2)` Where [.] denotes greatest integer function less than or equal to x]

Explanation

To solve the problem, we need to find the domain of the function \( f(x) = \frac{\lfloor x + \frac{1}{2} \rfloor}{\lfloor x - \frac{1}{2} \rfloor} \) such that \( f(x) \) is a prime number. Here, \( \lfloor . \rfloor \) denotes the greatest integer function.

### Step-by-step Solution:

1. **Understanding the Function**:
- The function is defined as \( f(x) = \frac{\lfloor x + \frac{1}{2} \rfloor}{\lfloor x - \frac{1}{2} \rfloor} \).
- We need to find when this function yields a prime number.

2. **Analyzing the Greatest Integer Function**:
- The expression \( \lfloor x + \frac{1}{2} \rfloor \) can be simplified:
- If \( x \) is in the interval \( [n - \frac{1}{2}, n + \frac{1}{2}) \) for some integer \( n \), then \( \lfloor x + \frac{1}{2} \rfloor = n \).
- Similarly, \( \lfloor x - \frac{1}{2} \rfloor \):
- If \( x \) is in the interval \( [n - \frac{1}{2}, n + \frac{1}{2}) \), then \( \lfloor x - \frac{1}{2} \rfloor = n - 1 \).

3. **Finding the Values of \( f(x) \)**:
- Thus, for \( x \) in \( [n - \frac{1}{2}, n + \frac{1}{2}) \):
\[ f(x) = \frac{n}{n - 1} \]
- This expression is defined for \( n > 1 \) (since \( n - 1 \) must not be zero).

4. **Finding Prime Values**:
- We need \( \frac{n}{n - 1} \) to be prime:
\[ \frac{n}{n - 1} = 1 + \frac{1}{n - 1} \]
- For this to be an integer, \( n - 1 \) must divide 1, which means \( n - 1 = 1 \) or \( n - 1 = -1 \).
- Hence, \( n = 2 \) (since \( n \) must be positive).

5. **Finding the Domain**:
- For \( n = 2 \):
\[ f(x) = \frac{2}{1} = 2 \]
- The corresponding interval for \( n = 2 \) is:
\[ [\frac{3}{2}, \frac{5}{2}) \]
- Thus, \( x_1 = \frac{3}{2} \) and \( x_2 = \frac{5}{2} \).

6. **Calculating the Required Expression**:
- We need to find \( 2(x_1^2 + x_2^2) \):
\[ x_1^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}, \quad x_2^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} \]
\[ x_1^2 + x_2^2 = \frac{9}{4} + \frac{25}{4} = \frac{34}{4} = \frac{17}{2} \]
\[ 2(x_1^2 + x_2^2) = 2 \cdot \frac{17}{2} = 17 \]

### Final Answer:
The value of \( 2(x_1^2 + x_2^2) \) is \( \boxed{17} \).