The domain of function `f(x)=(cos^(-1)x)/( [x]);` `[x]=GIF` is:

Options

A. `[-1,1]`
B. `[-1,1]-{0}`
C. `[-1,0) cup {1}`
D. `[-1, 0)`

Explanation

To find the domain of the function \( f(x) = \frac{\cos^{-1}(x)}{[x]} \), where \([x]\) is the greatest integer function (GIF), we need to consider the constraints imposed by both the numerator and the denominator.

### Step-by-Step Solution:

1. **Identify the domain of the numerator**:
The function \(\cos^{-1}(x)\) is defined for \(x\) in the interval \([-1, 1]\). Therefore, we have:
\[ -1 \leq x \leq 1 \]

2. **Identify the domain of the denominator**:
The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). The denominator cannot be zero, as division by zero is undefined. The value of \([x]\) is zero when \(x\) is in the interval \([0, 1)\). Therefore, we need to exclude this interval from the domain.

3. **Combine the constraints**:
From the first step, we have \(x \in [-1, 1]\). From the second step, we need to exclude the interval \([0, 1)\). Thus, we can express the valid intervals as:
\[ x \in [-1, 0) \cup [1, 1] \]

4. **Simplify the intervals**:
The interval \([1, 1]\) is just the single point \(1\). Therefore, the domain of \(f(x)\) can be written as:
\[ x \in [-1, 0) \cup \{1\} \]

### Final Domain:
The domain of the function \( f(x) = \frac{\cos^{-1}(x)}{[x]} \) is:
\[ [-1, 0) \cup \{1\} \]