What is the domain and range of y=sec x?

The function \( y = \sec x \) is defined as the reciprocal of the cosine function. To understand its domain and range, let’s break it down.

Domain of \( y = \sec x \)

The domain consists of all the values of \( x \) for which the function is defined. Since \( \sec x \) is the reciprocal of \( \cos x \), it is undefined wherever \( \cos x = 0 \). This occurs at:

• \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \)

Thus, the domain of \( y = \sec x \) can be expressed as:

Domain: \( x \in \mathbb{R} \) except \( x = \frac{\pi}{2} + n\pi \)

Range of \( y = \sec x \)

The range includes all possible output values of the function. Since \( \sec x \) is the reciprocal of \( \cos x \), it can take values greater than or equal to 1 or less than or equal to -1. Therefore, the range is:

Range: \( y \leq -1 \) or \( y \geq 1 \)

In summary, the function \( y = \sec x \) has a domain that excludes points where cosine is zero and a range that includes all values outside the interval (-1, 1).