Range of `f(x)=sin^(-1)x+tan^(-1)x+sec^(-1)x` is:

Options

A. `(pi/4,(3pi)/4)`
B. `[pi/4,(3pi)/4]`
C. `{pi/4,(3pi)/4}`
D. None of these

Answer

To find the range of the function \( f(x) = \sin^{-1}(x) + \tan^{-1}(x) + \sec^{-1}(x) \), we will analyze the individual components of the function and their respective ranges.

### Step 1: Identify the domains of each function
1. **Domain of \( \sin^{-1}(x) \)**: The domain is \( x \in [-1, 1] \) and the range is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
2. **Domain of \( \tan^{-1}(x) \)**: The domain is \( x \in (-\infty, \infty) \) and the range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
3. **Domain of \( \sec^{-1}(x) \)**: The domain is \( x \leq -1 \) or \( x \geq 1 \) and the range is \( [0, \pi] \) excluding \( \frac{\pi}{2} \).

### Step 2: Find the common domain
Since \( \sec^{-1}(x) \) restricts \( x \) to \( x \leq -1 \) or \( x \geq 1 \), we will evaluate \( f(x) \) at the endpoints of the domain of \( \sin^{-1}(x) \) which is \( x = -1 \) and \( x = 1 \).

### Step 3: Evaluate \( f(x) \) at \( x = 1 \)
\[ f(1) = \sin^{-1}(1) + \tan^{-1}(1) + \sec^{-1}(1) \]
- \( \sin^{-1}(1) = \frac{\pi}{2} \)
- \( \tan^{-1}(1) = \frac{\pi}{4} \)
- \( \sec^{-1}(1) = 0 \)

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

### Step 4: Evaluate \( f(x) \) at \( x = -1 \)
\[ f(-1) = \sin^{-1}(-1) + \tan^{-1}(-1) + \sec^{-1}(-1) \]
- \( \sin^{-1}(-1) = -\frac{\pi}{2} \)
- \( \tan^{-1}(-1) = -\frac{\pi}{4} \)
- \( \sec^{-1}(-1) = \pi \)

Thus,
\[ f(-1) = -\frac{\pi}{2} - \frac{\pi}{4} + \pi = -\frac{\pi}{2} - \frac{\pi}{4} + \frac{4\pi}{4} = \frac{\pi}{4} \]

### Step 5: Determine the range of \( f(x) \)
From our evaluations:
- At \( x = 1 \), \( f(1) = \frac{3\pi}{4} \)
- At \( x = -1 \), \( f(-1) = \frac{\pi}{4} \)

Since \( \tan^{-1}(x) \) is continuous and takes all values between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) as \( x \) varies over the real numbers, and since both \( \sin^{-1}(x) \) and \( \sec^{-1}(x) \) contribute to the overall function, we conclude that the range of \( f(x) \) is:
\[ \text{Range of } f(x) = \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \]

### Final Answer
The range of \( f(x) = \sin^{-1}(x) + \tan^{-1}(x) + \sec^{-1}(x) \) is \( \left[\frac{\pi}{4}, \frac{3\pi}{4}\right] \).

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