The domain of the function `f(x) = sqrt((4-x^(2))/([x]+2))` where [x] denotes the greatest integer less than or equal to x,is

Options

A. `[-1, 2]`
B. `(-oo, -2 )`
C. `(-oo, -2) uu [-1,2]`
D. none of these

Explanation

To find the domain of the function \( f(x) = \sqrt{\frac{4 - x^2}{[x] + 2}} \), we need to ensure that the expression inside the square root is non-negative and that the denominator is not zero. Here are the steps to find the domain:

### Step 1: Set the conditions for the square root
The expression inside the square root must be greater than or equal to zero:
\[ \frac{4 - x^2}{[x] + 2} \geq 0 \]
This leads to two conditions:
1. \( 4 - x^2 \geq 0 \)
2. \( [x] + 2 > 0 \)

### Step 2: Solve the first condition
From \( 4 - x^2 \geq 0 \):
\[ 4 \geq x^2 \implies -2 \leq x \leq 2 \]
This means \( x \) must lie in the interval \([-2, 2]\).

### Step 3: Solve the second condition
From \( [x] + 2 > 0 \):
\[ [x] > -2 \implies [x] \geq -1 \]
The greatest integer function \([x]\) takes integer values, so this means:
\[ [x] = -1, 0, 1, 2, \ldots \]
This implies:
\[ -1 \leq x < 0 \quad \text{or} \quad 0 \leq x < 1 \quad \text{or} \quad 1 \leq x < 2 \]

### Step 4: Combine the conditions
Now we need to combine the intervals from Step 2 and Step 3:
- From \( 4 - x^2 \geq 0 \), we have \([-2, 2]\).
- From \( [x] \geq -1 \), we have the intervals:
- \([-1, 0)\)
- \([0, 1)\)
- \([1, 2)\)

### Step 5: Determine the valid intervals
Now we will find the intersection of \([-2, 2]\) with the intervals from the second condition:
1. Intersection with \([-1, 0)\): \([-1, 0)\)
2. Intersection with \([0, 1)\): \([0, 1)\)
3. Intersection with \([1, 2)\): \([1, 2)\)

### Step 6: Exclude points where the denominator is zero
We also need to ensure that the denominator \([x] + 2 \neq 0\):
\[ [x] \neq -2 \implies [x] \text{ can be } -1, 0, 1, 2, \ldots \]
Since \([x] = -2\) is not possible for any \(x\) in our intervals, we do not need to exclude any additional points.

### Final Domain
Combining all valid intervals, we have:
\[ [-2, -1) \cup [0, 1) \cup [1, 2) \]

### Conclusion
Thus, the domain of the function \( f(x) \) is:
\[ [-2, -1) \cup [0, 2) \]