Problem 51 - Precalculus with Calculus Previews

Question

One definition of the arcsecant function is \(y=\) \(\operatorname{arcsec} x\) (or \(\left.y=\sec ^{-1} x\right)\) if and only if \(x=\sec y\) where \(0 \leq y \leq \pi\) and \(y \neq \pi / 2\). (See Problem 52 for an alternative definition.) Graph \(y=\operatorname{arcsec} x,\) and give the domain and the range of this function.

Step-by-Step Solution

Answer

The domain of \(\operatorname{arcsec} x\) is \(( -\infty, -1 ] \cup [ 1, \infty )\), and the range is \([0, \pi/2) \cup (\pi/2, \pi]\).

Step 1: Understand the Definition of Arcsecant

The arcsecant function, denoted as \( y = \operatorname{arcsec} x \), is the inverse of the secant function. The relationship between \( x \) and \( y \) is given by \( x = \sec y \). \( y \) is restricted to the interval \( 0 \leq y \leq \pi \), excluding \( \pi/2 \). This ensures that the function remains one-to-one and hence has an inverse.

Step 2: Identify the Domain of Arcsecant

Since \( y = \arcsec x \) implies \( x = \sec y \), and the secant function is defined for all real numbers \( y \) except for odd multiples of \( \pi/2 \), the domain for \( x \) in the arcsecant function must fulfill \( |x| \geq 1 \). This means the domain for \( x \) is \(( -\infty, -1 ] \cup [ 1, \infty )\).

Step 3: Determine the Range of Arcsecant

Given the restriction \( 0 \leq y \leq \pi \) and \( y eq \pi/2 \), the arcsecant function will output values within this interval except for \( \pi/2 \). Hence, the range is \([0, \pi/2) \cup (\pi/2, \pi]\).

Step 4: Graph the Arcsecant Function

To graph \( y = \operatorname{arcsec} x \), consider its definition based on the secant function. The graph will have two branches, one for negative values (mirror of secant) on \(( -\infty, -1 ]\) and one for positive values on \([1, \infty )\). The graph doesn't exist between \(-1\) and \(1\) because secant values are only less than -1 or greater than 1. There is a vertical asymptote at \(x = 0\) for \(y = \pi/2\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions provide us with the means to determine an angle given a trigonometric ratio. The arcsecant function, often denoted as \( y = \operatorname{arcsec} x \) or \( y = \sec^{-1} x \), specifically answers the question: "What angle \( y \) yields a secant of \( x \)?" However, to ensure that these functions are well-defined and unique (one-to-one), we limit the range of the angle \( y \) in these inverse functions.

The relationship \( x = \sec y \) for arcsecant comes with some restrictions to maintain this one-to-one correspondence. The common interval chosen for the arcsecant is \( [0, \pi] \), excluding \( \pi/2 \), because the secant function returns values for all real numbers except when \( y \) is an odd multiple of \( \pi/2 \), where it is undefined. Understanding these intricacies is crucial as they explain why inverse trigonometric functions appear in specific ways and help assess their feasibility in calculations.

Domain and Range

The domain and range of a function describe the set of possible input values (domain) and output values (range), respectively. For the arcsecant function, the domain comprises values of \( x \) where \( |x| \geq 1 \). This stems from the secant function \( x = \sec y \), where its values are either less than -1 or greater than 1, making it undefined for \(-1 < x < 1\).

Therefore, the arcsecant function's domain is \(( -\infty, -1 ] \cup [ 1, \infty )\). This mirrors the intervals where the secant is valid excluding its asymptotic values. For the range, since we are considering values of \( y \) in the interval \([0, \pi]\) but excluding \( \pi/2 \), we have the range as \([0, \pi/2) \cup (\pi/2, \pi]\). This results from the fact that such angles ensure \( x = \sec y \) lies within the allowed input values for arcsecant. Understanding these intervals helps clarify why certain values are excluded and how the graph behaves as a result.

Graphing Functions

Graphing functions like the arcsecant involves interpreting its behavior and plotting its values based on the function's characteristics. For \( y = \operatorname{arcsec} x \), the graph is constructed from two disjoint branches due to its domain \(( -\infty, -1 ] \cup [ 1, \infty )\). These branches occur because the secant function itself has two separate curves outside the interval of \(-1\) to \(1\).

On a graph, \( y = \operatorname{arcsec} x \) has a vertical asymptote at \( x = 0 \) because secant approaches \( \pm \infty \) as it nears \( \pi/2 \). When plotting, consider the behavior in both relevant intervals. For \( x \leq -1 \), the arcsecant reflects the mirrored properties of the secant function, translating to a decreasing curve. Similarly, for \( x \geq 1 \), it reflects an increasing curve. It’s vital to also note that there are no points between \(-1\) and \(1\), emphasizing why understanding the behavior of inverse trigonometric functions is as much about identifying gaps as it is about pinpointing specific trends.