`f(x)=1/sqrt([x]^(2)-[x]-6)`, where `[*]` denotes the greatest integer function.

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{[\![x]\!]^2 - [\![x]\!] - 6}} \), where \([\![x]\!]\) denotes the greatest integer function, we need to ensure that the expression inside the square root is positive, since the denominator cannot be zero or negative.

### Step 1: Set the condition for the square root
We start by setting the condition that the expression inside the square root must be greater than zero:
\[ [\![x]\!]^2 - [\![x]\!] - 6 > 0 \]

### Step 2: Substitute the greatest integer function
Let \( t = [\![x]\!] \). Then we rewrite the inequality:
\[ t^2 - t - 6 > 0 \]

### Step 3: Factor the quadratic expression
Next, we factor the quadratic:
\[ t^2 - t - 6 = (t - 3)(t + 2) \]
Thus, we have:
\[ (t - 3)(t + 2) > 0 \]

### Step 4: Determine the critical points
The critical points from the factors are \( t = 3 \) and \( t = -2 \). We will analyze the sign of the product in the intervals defined by these points:
- \( t < -2 \)
- \( -2 < t < 3 \)
- \( t > 3 \)

### Step 5: Test the intervals
1. For \( t < -2 \) (e.g., \( t = -3 \)):
\[ (-3 - 3)(-3 + 2) = (-6)(-1) = 6 > 0 \quad \text{(positive)} \]
2. For \( -2 < t < 3 \) (e.g., \( t = 0 \)):
\[ (0 - 3)(0 + 2) = (-3)(2) = -6 < 0 \quad \text{(negative)} \]
3. For \( t > 3 \) (e.g., \( t = 4 \)):
\[ (4 - 3)(4 + 2) = (1)(6) = 6 > 0 \quad \text{(positive)} \]

### Step 6: Combine the intervals
The solution to the inequality \( (t - 3)(t + 2) > 0 \) is:
\[ t < -2 \quad \text{or} \quad t > 3 \]
Thus, we can express this in interval notation as:
\[ (-\infty, -2) \cup (3, \infty) \]

### Step 7: Translate back to \( x \)
Since \( t = [\![x]\!] \), we need to find the corresponding \( x \) values:
- For \( t < -2 \): The greatest integer function outputs integers less than or equal to -3, which corresponds to \( x < -2 \).
- For \( t > 3 \): The greatest integer function outputs integers greater than or equal to 4, which corresponds to \( x \geq 4 \).

### Final Domain
Thus, the domain of \( f(x) \) is:
\[ (-\infty, -2) \cup [4, \infty) \]