91Ó°ÊÓ

The secant function, denoted as \( \sec(x) \), is an important trigonometric function defined as the reciprocal of the cosine function. To understand the domain of the secant function, consider its definition: \( \sec(x) = \frac{1}{\cos(x)} \). Here, the function is only defined where the cosine function does not equal zero, because division by zero is undefined.

Hence, we must exclude values where \( \cos(x) = 0 \), which occur at \( x = 2n\pi \pm \frac{\pi}{2} \), where \( n \) is any integer. Therefore, the domain of the secant function is all real numbers except these values. In mathematical terms, it's written as:

• \( x \in \mathbb{R} \setminus \{2n\pi \pm \frac{\pi}{2}\} \) for \( n \in \mathbb{Z} \)

Once we've understood what the secant function is and where it's undefined, we can explore its range. Since \( \sec(x) = \frac{1}{\cos(x)} \), and given that the cosine function ranges between \(-1\) and \(1\), the secant function interestingly lacks values between \(-1\) and \(1\) in its range.

Thus, the range of the secant function includes all real numbers except those within the open interval \((-1, 1)\). In interval notation, the range of \( \sec(x) \) is denoted as:

• \((-\infty, -1] \cup [1, \infty)\)

Interval notation is a mathematical notation used to describe a range of values in a concise way. It is particularly useful when dealing with functions like the secant and its inverse,\( \sec^{-1}(x) \), where domain and range intervals need to be well expressed.

An interval includes its endpoints if it is 'closed', denoted by square brackets like \([a, b]\), indicating that both \(a\) and \(b\) are included. Conversely, an interval excludes its endpoints if it is 'open', represented by parentheses like \((a, b)\), excluding \(a\) and \(b\). You can also have half-open intervals, such as \([a, b)\), where \(a\) is included but \(b\) is not.

Using interval notation makes it easy to express ranges clearly, such as with the secant function where the range \((-\infty, -1]\cup[1, \infty)\) clearly shows the values the function can take. Understanding interval notation will help you accurately interpret and describe mathematical domains and ranges.