Problem 52 - Precalculus with Calculus Previews

Question

An alternative definition of the arcsecant function can be made by restricting the domain of the secant function to \([0, \pi / 2) \cup[\pi, 3 \pi / 2) .\) Under this restriction, define the arcsecant function. Graph \(y=\operatorname{arcsec} x,\) and give the domain and the range of this function.

Step-by-Step Solution

Answer

The domain is \((-\infty, -1]\cup [1, \infty)\) and the range is \([0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\).

Step 1: Understanding Arcsecant Definition

The arcsecant function, denoted as \(y = \operatorname{arcsec} x\), is the inverse of the secant function. For \(y = \operatorname{arcsec} x\), this means \(x = \sec y\) and \(y\) is defined based on specific intervals of \([0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\). These intervals exclude values where the secant is undefined or negative.

Step 2: Determining the Domain

Since secant is defined as \(\sec y = \frac{1}{\cos y}\), \(x\) must not be in [-1,1], the range of cosine when \(y\) lies in \([0, \frac{\pi}{2})\) and \( [\pi, \frac{3\pi}{2})\). Therefore, the domain of \(y = \operatorname{arcsec} x\) is \(x \in (-\infty, -1]\cup [1, \infty)\).

Step 3: Determining the Range

For the arcsecant function, since \(x = \sec y\), \(y\) must lie within the intervals where \(\sec y\) can invert to reach its domain \((-\infty, -1] \cup [1, \infty)\). Therefore, the range is \(y \in [0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\).

Step 4: Graphing the Function

To graph \(y = \operatorname{arcsec} x\), plot the function with the domain \((-\infty, -1]\cup [1, \infty)\) on the x-axis and the range \([0, \frac{\pi}{2}) \cup[\pi, \frac{3\pi}{2})\) on the y-axis. The graph shows the reflection over the line \(y = x\) from the restriction of the secant function.

Step 5: Summary of Domain and Range

The function \(y = \operatorname{arcsec} x\) has a domain of \((-\infty, -1]\cup [1, \infty)\), and the range is \([0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\). The graph consists of two separate branches corresponding to these defined ranges.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are crucial because they help us find angles when the value of a trigonometric function is known. These functions essentially reverse the operations of the standard trigonometric functions. For example, the arcsecant function, denoted as \(y = \operatorname{arcsec} x\), is the inverse of the secant function. It tells us which angle corresponds to a given secant. In this reversal process, the values become particular subsets of angles which make the inverse valid and practical for calculations. The secant function, being the reciprocal of the cosine function, only has an inverse when carefully limited to certain intervals; thus, the arcsecant is likewise restricted.

Domain and Range

The domain and range are critical to understanding any function, especially inverse functions like arcsecant. The domain of \(y = \operatorname{arcsec} x\) is derived from its relation with secant, noting where secant values were originally undefined or negative. Since \(\sec y = \frac{1}{\cos y}\), secant, and thus arcsecant, isn't defined for \([-1,1]\) due to cosine's behavior. Therefore, the domain of arcsecant comprises two intervals: \((-fty, -1]\) and \([1, \infty)\). On the other hand, the range of \(y = \operatorname{arcsec} x\) is given by where the secant function's inverse can map its domain, specifically \([0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\). These intervals arise from how inverse functions reflect over the line \(y = x\), realigning the secant function's outputs back to valid angle measurements.

Graphing Functions

When graphing \(y = \operatorname{arcsec} x\), visualization aids in comprehending the function's unique behavior through its allowable domain and range. Begin by plotting the x-axis covering \((-fty, -1]\) and \([1, \infty)\), which comprehensively represents where the arcsecant function applies. Next, the y-axis represents the range \([0, \frac{\pi}{2}) \cup [\pi, \frac{3\pi}{2})\), showcasing where \(y\) values make the function results valid. The distinct portrayal over these ranges suggests the reflection of secant's restricted operations over \(y = x\). Through graphical depiction, the arcsecant function unveils its separate branches, each a testament to its inverse secant relationship. This visualization not only pushes the conceptual understanding of inverses but also reinforces how domain, range, and functional realignment collaborate in trigonometry.