The domain and range of `f(x) = sin^1x + cos^-1 x +tan^-1x + cot^-1x + sec^-1x + cosec^-1x` respectively are

Options

A. `{-1,1},(3pi)/2`
B. `{-1,1},pi/2`
C. `(-1,1),pi/2`
D. `(-1,1),2pi`

Answer

To find the domain and range of the function \( f(x) = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x + \cot^{-1}x + \sec^{-1}x + \csc^{-1}x \), we will analyze each component function separately.

### Step 1: Determine the domain of each component function.

1. **Domain of \( \sin^{-1}x \)**:
- The domain is \( x \in [-1, 1] \).

2. **Domain of \( \cos^{-1}x \)**:
- The domain is \( x \in [-1, 1] \).

3. **Domain of \( \tan^{-1}x \)**:
- The domain is \( x \in (-\infty, \infty) \).

4. **Domain of \( \cot^{-1}x \)**:
- The domain is \( x \in (-\infty, \infty) \).

5. **Domain of \( \sec^{-1}x \)**:
- The domain is \( x \in (-\infty, -1] \cup [1, \infty) \).

6. **Domain of \( \csc^{-1}x \)**:
- The domain is \( x \in (-\infty, -1] \cup [1, \infty) \).

### Step 2: Find the intersection of the domains.

To find the overall domain of \( f(x) \), we take the intersection of all the individual domains:

- The common domain for \( \sin^{-1}x \) and \( \cos^{-1}x \) is \( [-1, 1] \).
- The domains of \( \tan^{-1}x \) and \( \cot^{-1}x \) do not restrict the values further since they are defined for all real numbers.
- The domains of \( \sec^{-1}x \) and \( \csc^{-1}x \) restrict the values to \( (-\infty, -1] \cup [1, \infty) \).

Thus, the intersection of these domains is:
- For \( x \in [-1, 1] \) and \( x \in (-\infty, -1] \cup [1, \infty) \), the only values that are common are \( -1 \) and \( 1 \).

### Conclusion for Domain:
The domain of \( f(x) \) is \( \{-1, 1\} \).

### Step 3: Determine the range of \( f(x) \).

Next, we will find the range of \( f(x) \):

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

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

### Conclusion for Range:
The range of \( f(x) \) is \( \{-\frac{\pi}{2}, \frac{3\pi}{2}\} \).

### Final Answer:
The domain and range of \( f(x) \) are:
- Domain: \( \{-1, 1\} \)
- Range: \( \{-\frac{\pi}{2}, \frac{3\pi}{2}\} \)