The domain of definition of the function `f(x)=tan((pi)/([x+2]))`, is where [] represents greatest integer function less than or equal to x.

Options

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

Answer

To find the domain of the function \( f(x) = \tan\left(\frac{\pi}{[x+2]}\right) \), where \([x]\) denotes the greatest integer function (also known as the floor function), we need to determine the values of \( x \) for which the function is defined.

### Step-by-Step Solution:

1. **Identify the Conditions for the Tangent Function**:
The tangent function is defined for all real numbers except where its argument is an odd multiple of \( \frac{\pi}{2} \). This means we need to ensure that:
\[ \frac{\pi}{[x+2]} \neq \frac{\pi}{2} + n\pi \quad \text{for any integer } n \]
Simplifying this gives:
\[ [x+2] \neq \frac{2}{1 + 2n} \quad \text{for any integer } n \]

2. **Determine the Values of \([x+2]\)**:
Since \([x+2]\) is an integer, we need to find integer values that are not equal to \( \pm 2 \) (as derived from the previous step). Thus:
\[ [x+2] \neq 2 \quad \text{and} \quad [x+2] \neq -2 \]

3. **Translate Conditions into Inequalities**:
- For \([x+2] \neq 2\):
\[ x + 2 \neq 2 \implies x \neq 0 \]
- For \([x+2] \neq -2\):
\[ x + 2 \neq -2 \implies x \neq -4 \]

4. **Consider the Greatest Integer Function**:
The greatest integer function \([x+2]\) can take values depending on the range of \( x \):
- If \( [x+2] = k \) (where \( k \) is an integer), then:
\[ k \leq x + 2 < k + 1 \implies k - 2 \leq x < k - 1 \]
- Therefore, we need to exclude intervals where \( [x+2] = 2 \) or \( [x+2] = -2 \).

5. **Identify the Excluded Intervals**:
- For \( [x+2] = 2 \):
\[ 0 \leq x < 1 \]
- For \( [x+2] = -2 \):
\[ -4 \leq x < -3 \]

6. **Combine the Conditions**:
The domain of \( f(x) \) is all real numbers except the intervals derived from the conditions above:
\[ \text{Domain of } f(x) = \mathbb{R} \setminus \left( [-4, -3) \cup [0, 1) \right) \]

### Final Answer:
Thus, the domain of the function \( f(x) = \tan\left(\frac{\pi}{[x+2]}\right) \) is:
\[ \mathbb{R} \setminus \left( [-4, -3) \cup [0, 1) \right) \]