Determine the domain and range of sec^-1x.

The function \( \sec^{-1}(x) \), also known as the inverse secant function, has specific characteristics that define its domain and range.

Domain of \( \sec^{-1}(x) \)

The domain refers to the set of values that \( x \) can take. For the inverse secant function, the domain is limited to:

• \( x \leq -1 \)
• \( x \geq 1 \)

This means that \( x \) can be any number less than or equal to -1 or greater than or equal to 1.

Range of \( \sec^{-1}(x) \)

The range indicates the possible output values of the function. For \( \sec^{-1}(x) \), the range is:

• \( 0 \leq y < \frac{\pi}{2} \)
• \( \frac{\pi}{2} < y \leq \pi \)

This means that the output values can be any angle from 0 to \( \pi \), excluding \( \frac{\pi}{2} \).

Summary

In summary, the domain of \( \sec^{-1}(x) \) is \( (-\infty, -1] \cup [1, \infty) \), and the range is \( [0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi] \).