`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 (also known as the floor function), we need to ensure that the expression under the square root is positive. Let's solve this step by step.

### Step 1: Set up the inequality
We need to ensure that the expression inside the square root is greater than zero:
\[ [x]^2 - [x] - 6 > 0 \]

### Step 2: Substitute \( y \) for \([x]\)
Let \( y = [x] \). Then we can rewrite the inequality as:
\[ y^2 - y - 6 > 0 \]

### Step 3: Factor the quadratic expression
To factor the quadratic, we look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). Thus, we can factor the expression:
\[ (y - 3)(y + 2) > 0 \]

### Step 4: Determine the critical points
The critical points from the factors are:
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

### Step 5: Test intervals
We will test the intervals determined by the critical points \( y = -2 \) and \( y = 3 \):
1. \( (-\infty, -2) \)
2. \( (-2, 3) \)
3. \( (3, \infty) \)

- For \( y < -2 \) (e.g., \( y = -3 \)):
\[ (-3 - 3)(-3 + 2) = (-6)(-1) = 6 > 0 \quad \text{(satisfied)} \]

- For \( -2 < y < 3 \) (e.g., \( y = 0 \)):
\[ (0 - 3)(0 + 2) = (-3)(2) = -6 < 0 \quad \text{(not satisfied)} \]

- For \( y > 3 \) (e.g., \( y = 4 \)):
\[ (4 - 3)(4 + 2) = (1)(6) = 6 > 0 \quad \text{(satisfied)} \]

### Step 6: Combine the intervals
From our testing, we find that the inequality \( (y - 3)(y + 2) > 0 \) is satisfied in the intervals:
\[ y \in (-\infty, -2) \cup (3, \infty) \]

### Step 7: Translate back to \( x \)
Since \( y = [x] \), we need to find the corresponding intervals for \( x \):
- For \( y < -2 \): This means \( [x] \leq -3 \), which corresponds to \( x < -2 \).
- For \( y > 3 \): This means \( [x] \geq 4 \), which corresponds to \( x \geq 4 \).

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