Domain of `f(x)=cos^(-1)x+cot^(-1)x+cosec^(-1)x` is

Options

A. `[-1,1]`
B. `R`
C. `(-oo,-1]uu[1,oo)`
D. `{-1,1}`

Explanation

To find the domain of the function \( f(x) = \cos^{-1}x + \cot^{-1}x + \csc^{-1}x \), we need to determine the individual domains of each of the component functions and then find the intersection of these domains.

### Step-by-step Solution:

1. **Identify the domains of each function:**
- The domain of \( \cos^{-1}x \) is \( x \in [-1, 1] \).
- The domain of \( \cot^{-1}x \) is \( x \in \mathbb{R} \) (all real numbers).
- The domain of \( \csc^{-1}x \) is \( x \in (-\infty, -1] \cup [1, \infty) \) (all real numbers except the interval \((-1, 1)\)).

2. **Combine the domains:**
- We need to find the intersection of the three domains:
- For \( \cos^{-1}x \): \( [-1, 1] \)
- For \( \cot^{-1}x \): \( \mathbb{R} \)
- For \( \csc^{-1}x \): \( (-\infty, -1] \cup [1, \infty) \)

3. **Determine the intersection:**
- The intersection of \( [-1, 1] \) and \( (-\infty, -1] \cup [1, \infty) \) is:
- The only points that are common in both sets are \( -1 \) and \( 1 \).
- Therefore, the intersection is \( \{-1, 1\} \).

4. **Final domain:**
- Since \( \cot^{-1}x \) is defined for all real numbers, it does not restrict the intersection further.
- Thus, the domain of \( f(x) \) is \( x = -1 \) and \( x = 1 \).

### Conclusion:
The domain of the function \( f(x) = \cos^{-1}x + \cot^{-1}x + \csc^{-1}x \) is:
\[ \text{Domain of } f(x) = \{-1, 1\} \]