Let `f(x)=sec^-1(x-10)+cos^-1(10-x).` Then range of `f(x)` is

Options

A. `{0,pi/2,pi}`
B. `{0,pi/2}`
C. `{pi/2}`
D. `{pi}`

Answer

To find the range of the function \( f(x) = \sec^{-1}(x - 10) + \cos^{-1}(10 - x) \), we will analyze the individual components of the function.

### Step 1: Determine the domain of \( \sec^{-1}(x - 10) \)

The function \( \sec^{-1}(y) \) is defined for \( y \leq -1 \) or \( y \geq 1 \). Therefore, for \( \sec^{-1}(x - 10) \):

\[ x - 10 \leq -1 \quad \text{or} \quad x - 10 \geq 1 \]

Solving these inequalities:

1. \( x - 10 \leq -1 \)
\[ x \leq 9 \]

2. \( x - 10 \geq 1 \)
\[ x \geq 11 \]

Thus, the domain for \( \sec^{-1}(x - 10) \) is \( (-\infty, 9] \cup [11, \infty) \).

### Step 2: Determine the domain of \( \cos^{-1}(10 - x) \)

The function \( \cos^{-1}(y) \) is defined for \( 0 \leq y \leq 1 \). Therefore, for \( \cos^{-1}(10 - x) \):

\[ 0 \leq 10 - x \leq 1 \]

Solving these inequalities:

1. \( 10 - x \geq 0 \)
\[ x \leq 10 \]

2. \( 10 - x \leq 1 \)
\[ x \geq 9 \]

Thus, the domain for \( \cos^{-1}(10 - x) \) is \( [9, 10] \).

### Step 3: Find the intersection of the domains

The overall domain of \( f(x) \) is the intersection of the domains of \( \sec^{-1}(x - 10) \) and \( \cos^{-1}(10 - x) \):

\[ (-\infty, 9] \cup [11, \infty) \cap [9, 10] = [9, 10] \]

### Step 4: Evaluate \( f(x) \) at the endpoints of the domain

Now we will evaluate \( f(x) \) at the endpoints \( x = 9 \) and \( x = 10 \).

1. **At \( x = 9 \)**:
\[ f(9) = \sec^{-1}(9 - 10) + \cos^{-1}(10 - 9) = \sec^{-1}(-1) + \cos^{-1}(1) \]
\[ = \pi + 0 = \pi \]

2. **At \( x = 10 \)**:
\[ f(10) = \sec^{-1}(10 - 10) + \cos^{-1}(10 - 10) = \sec^{-1}(0) + \cos^{-1}(0) \]
\[ = \text{undefined} + \frac{\pi}{2} \quad \text{(since } \sec^{-1}(0) \text{ is undefined)} \]

### Step 5: Determine the range of \( f(x) \)

Since \( f(x) \) is defined only at \( x = 9 \) and is undefined at \( x = 10 \), the only value we have is:

\[ f(9) = \pi \]

Thus, the range of \( f(x) \) is:

\[ \text{Range of } f(x) = \{ \pi \} \]

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

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