The domain (प्रांत ) 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 \([x]\) denotes the greatest integer function (GIF), we need to analyze the restrictions imposed by both the numerator and the denominator.

### Step 1: Determine the domain of \( \cos^{-1}(x) \)

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

**Hint:** The inverse cosine function is only defined for values between -1 and 1, inclusive.

### Step 2: Determine the restrictions from the greatest integer function \([x]\)

The greatest integer function \([x]\) gives the largest integer less than or equal to \( x \). We need to ensure that \([x] \neq 0\) because it is in the denominator. The greatest integer function \([x]\) equals 0 when \( x \) is in the interval \([0, 1)\). Therefore, we must exclude this interval from our domain:
\[ x \notin [0, 1) \]

**Hint:** The greatest integer function equals zero for values in the interval [0, 1), so we must exclude this range.

### Step 3: Combine the restrictions

From Step 1, we have \( -1 \leq x \leq 1 \). From Step 2, we exclude the interval \([0, 1)\). Thus, we need to find the intersection of these two conditions.

The valid intervals for \( x \) are:
1. From \(-1\) to \(0\) (inclusive of -1, exclusive of 0).
2. The point \(1\) (since \(\cos^{-1}(1) = 0\) is valid).

Thus, the combined domain is:
\[ [-1, 0) \cup \{1\} \]

### Final Domain

The final domain of the function \( f(x) \) is:
\[ [-1, 0) \cup \{1\} \]

### Conclusion

Thus, the correct option is:
**Option C:** \([-1, 0) \cup \{1\}\)