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 \( f(x) = \frac{1}{\sqrt{[\![x]\!]^2 - [\![x]\!] - 20}} \), where \([\![x]\!]\) represents the greatest integer function, we need to ensure that the expression inside the square root is positive. This means we need to solve the inequality:

\[ [\![x]\!]^2 - [\![x]\!] - 20 > 0 \]

### Step 1: Set up the quadratic inequality

The quadratic can be expressed as:

\[ t^2 - t - 20 > 0 \]

where \( t = [\![x]\!] \).

### Step 2: Factor the quadratic

Next, we factor the quadratic:

\[ t^2 - t - 20 = (t - 5)(t + 4) \]

### Step 3: Find the critical points

Setting the factors equal to zero gives us the critical points:

\[ t - 5 = 0 \quad \Rightarrow \quad t = 5 \]
\[ t + 4 = 0 \quad \Rightarrow \quad t = -4 \]

### Step 4: Analyze the intervals

We need to determine where the product \((t - 5)(t + 4) > 0\). The critical points divide the number line into intervals:

1. \( (-\infty, -4) \)
2. \( (-4, 5) \)
3. \( (5, \infty) \)

### Step 5: Test the intervals

- For \( t < -4 \) (e.g., \( t = -5 \)):
\[ (-5 - 5)(-5 + 4) = (-10)(-1) > 0 \quad \text{(True)} \]

- For \( -4 < t < 5 \) (e.g., \( t = 0 \)):
\[ (0 - 5)(0 + 4) = (-5)(4) < 0 \quad \text{(False)} \]

- For \( t > 5 \) (e.g., \( t = 6 \)):
\[ (6 - 5)(6 + 4) = (1)(10) > 0 \quad \text{(True)} \]

### Step 6: Combine the results

The solution to the inequality is:

\[ t \in (-\infty, -4) \cup (5, \infty) \]

### Step 7: Translate back to \( x \)

Since \( t = [\![x]\!] \), we need to find the corresponding \( x \) values:

- For \( t < -4 \):
\[ [\![x]\!] \leq -5 \quad \Rightarrow \quad x < -4 \]

- For \( t > 5 \):
\[ [\![x]\!] \geq 6 \quad \Rightarrow \quad x \geq 6 \]

### Final Domain

Thus, the domain of the function \( f(x) \) is:

\[ x \in (-\infty, -4) \cup [6, \infty) \]