Range of the function
`f(x)=cos^(-1)x+2cot^(-1)x+3cosec^(-1)x` is equal to

Options

A. `[pi,2pi]`
B. `{pi,2pi}`
C. `{2pi,3pi}`
D. `[2pi,3pi]`

Explanation

To find the range of the function \( f(x) = \cos^{-1} x + 2 \cot^{-1} x + 3 \csc^{-1} x \), we will follow these steps:

### Step 1: Determine the domains of the individual functions
1. **Domain of \( \cos^{-1} x \)**: The domain is \( x \in [-1, 1] \).
2. **Domain of \( \cot^{-1} x \)**: The domain is \( x \in \mathbb{R} \) (all real numbers).
3. **Domain of \( \csc^{-1} x \)**: The domain is \( x \in (-\infty, -1] \cup [1, \infty) \).

### Step 2: Find the overall domain of \( f(x) \)
To find the overall domain of \( f(x) \), we need to take the intersection of the domains of the individual functions:
- The intersection of \( [-1, 1] \) (from \( \cos^{-1} x \)) and \( (-\infty, -1] \cup [1, \infty) \) (from \( \csc^{-1} x \)) gives us the points where both functions are defined.
- The only points that satisfy both conditions are \( x = -1 \) and \( x = 1 \).

Thus, the domain of \( f(x) \) is \( \{-1, 1\} \).

### Step 3: Calculate \( f(-1) \) and \( f(1) \)
1. **Calculate \( f(-1) \)**:
\[ f(-1) = \cos^{-1}(-1) + 2 \cot^{-1}(-1) + 3 \csc^{-1}(-1) \]
- \( \cos^{-1}(-1) = \pi \)
- \( \cot^{-1}(-1) = \frac{3\pi}{4} \) (since \( \cot^{-1}(-1) = \frac{\pi}{4} + \pi \))
- \( \csc^{-1}(-1) = -\frac{\pi}{2} \)

Therefore,
\[ f(-1) = \pi + 2 \left(-\frac{\pi}{4}\right) + 3 \left(-\frac{\pi}{2}\right) = \pi - \frac{\pi}{2} - \frac{3\pi}{2} = \pi \]

2. **Calculate \( f(1) \)**:
\[ f(1) = \cos^{-1}(1) + 2 \cot^{-1}(1) + 3 \csc^{-1}(1) \]
- \( \cos^{-1}(1) = 0 \)
- \( \cot^{-1}(1) = \frac{\pi}{4} \)
- \( \csc^{-1}(1) = \frac{\pi}{2} \)

Therefore,
\[ f(1) = 0 + 2 \left(\frac{\pi}{4}\right) + 3 \left(\frac{\pi}{2}\right) = \frac{\pi}{2} + \frac{3\pi}{2} = 2\pi \]

### Step 4: Determine the range of \( f(x) \)
The values we calculated are:
- \( f(-1) = \pi \)
- \( f(1) = 2\pi \)

Thus, the range of the function \( f(x) \) is \( \{ \pi, 2\pi \} \).

### Final Answer
The range of the function \( f(x) \) is \( \{ \pi, 2\pi \} \).

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