Domain of `sec^(-1)x` is :

Options

A. R-(-1,1)
B. R
C. `[-1,1]`
D. `R-(0,1)`

Answer

To find the domain of the function \( \sec^{-1}(x) \), we need to understand the range of the secant function, \( \sec(x) \).

### Step-by-Step Solution:

1. **Understanding the Secant Function**:
The secant function is defined as:
\[ \sec(x) = \frac{1}{\cos(x)} \]
The secant function is undefined wherever \( \cos(x) = 0 \), which occurs at odd multiples of \( \frac{\pi}{2} \).

2. **Range of the Secant Function**:
The range of \( \sec(x) \) can be determined from the behavior of the cosine function. The cosine function oscillates between -1 and 1. Therefore, the secant function, which is the reciprocal of the cosine function, will take values outside this interval:
\[ \text{Range of } \sec(x) = (-\infty, -1] \cup [1, \infty) \]

3. **Domain of the Inverse Secant Function**:
The domain of the inverse function \( \sec^{-1}(x) \) is the same as the range of the secant function. Thus, the domain of \( \sec^{-1}(x) \) is:
\[ \text{Domain of } \sec^{-1}(x) = (-\infty, -1] \cup [1, \infty) \]

4. **Conclusion**:
Therefore, the domain of \( \sec^{-1}(x) \) is:
\[ \boxed{(-\infty, -1] \cup [1, \infty)} \]