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 analyze the domain of each individual component function.

### Step 1: Determine the domain of \( \cos^{-1}x \)
The function \( \cos^{-1}x \) is defined for:
\[ -1 \leq x \leq 1 \]
So, the domain of \( \cos^{-1}x \) is \( [-1, 1] \).

### Step 2: Determine the domain of \( \cot^{-1}x \)
The function \( \cot^{-1}x \) is defined for all real numbers:
\[ x \in \mathbb{R} \]
So, the domain of \( \cot^{-1}x \) is \( (-\infty, \infty) \).

### Step 3: Determine the domain of \( \csc^{-1}x \)
The function \( \csc^{-1}x \) is defined for:
\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]
So, the domain of \( \csc^{-1}x \) is \( (-\infty, -1] \cup [1, \infty) \).

### Step 4: Find the intersection of the domains
Now, we need to find the intersection of the domains of all three functions:
1. Domain of \( \cos^{-1}x \): \( [-1, 1] \)
2. Domain of \( \cot^{-1}x \): \( (-\infty, \infty) \)
3. Domain of \( \csc^{-1}x \): \( (-\infty, -1] \cup [1, \infty) \)

The intersection of these domains will give us the overall domain of \( f(x) \).

- The interval \( [-1, 1] \) from \( \cos^{-1}x \) intersects with \( (-\infty, -1] \) at the point \( -1 \).
- The interval \( [-1, 1] \) also intersects with \( [1, \infty) \) at the point \( 1 \).

Thus, the overall domain of \( f(x) \) is:
\[ \{-1, 1\} \]

### Final Answer
The domain of \( f(x) = \cos^{-1}x + \cot^{-1}x + \csc^{-1}x \) is \( \{-1, 1\} \).

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