The domain of the function `f(x)=(1)/(sqrt([x]^(2)-[x]-20))` is (where, `[.]` represents the greatest integer function)

Options

A. `(-oo, -4)uu[6, oo)`
B. `(-oo, 4]uu[6, oo)`
C. `(-oo, 4)uu(6, oo)`
D. None of these

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{[\lfloor x \rfloor]^2 - [\lfloor x \rfloor] - 20}} \), where \( [\cdot] \) denotes the greatest integer function (floor function), we need to ensure that the expression inside the square root is positive, as it is in the denominator.

### Step-by-Step Solution:

1. **Identify the expression inside the square root**:
\[ t = [\lfloor x \rfloor]^2 - [\lfloor x \rfloor] - 20 \]
We need \( t > 0 \).

2. **Let \( \alpha = [\lfloor x \rfloor] \)**:
The inequality becomes:
\[ \alpha^2 - \alpha - 20 > 0 \]

3. **Factor the quadratic**:
To factor \( \alpha^2 - \alpha - 20 \), we look for two numbers that multiply to \(-20\) and add to \(-1\). These numbers are \( -5 \) and \( 4 \). Thus, we can factor it as:
\[ (\alpha - 5)(\alpha + 4) > 0 \]

4. **Find the critical points**:
The critical points from the factors are:
\[ \alpha = 5 \quad \text{and} \quad \alpha = -4 \]

5. **Test the intervals**:
We will test the intervals determined by the critical points:
- For \( \alpha < -4 \): Choose \( \alpha = -5 \):
\[ (-5 - 5)(-5 + 4) = (-10)(-1) > 0 \quad \text{(positive)} \]
- For \( -4 < \alpha < 5 \): Choose \( \alpha = 0 \):
\[ (0 - 5)(0 + 4) = (-5)(4) < 0 \quad \text{(negative)} \]
- For \( \alpha > 5 \): Choose \( \alpha = 6 \):
\[ (6 - 5)(6 + 4) = (1)(10) > 0 \quad \text{(positive)} \]

6. **Determine the valid intervals**:
The function is positive for:
\[ \alpha < -4 \quad \text{or} \quad \alpha > 5 \]
Thus, the valid intervals for \( \alpha \) are:
\[ (-\infty, -4) \quad \text{and} \quad (5, \infty) \]

7. **Translate back to \( x \)**:
Since \( \alpha = [\lfloor x \rfloor] \), we need to find the corresponding \( x \) values:
- For \( \alpha < -4 \): This means \( [\lfloor x \rfloor] \leq -5 \), so \( x < -4 \).
- For \( \alpha > 5 \): This means \( [\lfloor x \rfloor] \geq 6 \), so \( x \geq 6 \).

8. **Combine the intervals**:
Therefore, the domain of \( f(x) \) is:
\[ (-\infty, -4) \cup [6, \infty) \]

### Final Answer:
The domain of the function \( f(x) \) is:
\[ x \in (-\infty, -4) \cup [6, \infty) \]