Problem 35 - Algebra 2

Question

Identify all asymptotes to the graph of \(g(x)=\sec x\)

Step-by-Step Solution

Answer

The vertical asymptotes occur at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \); there are no horizontal or slant asymptotes.

Step 1: Identify Vertical Asymptotes

The function \( g(x) = \sec x \) is defined as \( g(x) = \frac{1}{\cos x} \). Vertical asymptotes occur where the denominator is zero. Therefore, we need to find the values of \( x \) for which \( \cos x = 0 \). This occurs at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

Step 2: Confirm the Behavior Near Vertical Asymptotes

Near the vertical asymptotes \( x = \frac{\pi}{2} + k\pi \), \( g(x) = \frac{1}{\cos x} \) approaches \( +\infty \) or \( -\infty \), confirming that these are indeed vertical asymptotes.

Step 3: Check for Horizontal Asymptotes

A horizontal asymptote exists if the function approaches a certain value as \( x \to \pm \infty \). For \( g(x) = \sec x \), since \( x \) cannot tend to \( \pm \infty \) without encountering the vertical asymptotes periodically, there are no horizontal asymptotes.

Step 4: Check for Slant Asymptotes

Slant asymptotes occur if \( f(x) = ax + b + \frac{c}{x} \). Since \( \sec x \) is periodic and does not behave as a polynomial at \( x \to \pm \infty \), it does not have slant asymptotes.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes arise in rational functions where the denominator equals zero, leading to undefined values in the graph. In the case of the function \( g(x) = \sec x = \frac{1}{\cos x} \), determining vertical asymptotes involves finding where the cosine function becomes zero. This occurs at points \( x = \frac{\pi}{2} + k\pi \), with \( k \) representing any integer. These points correspond to the vertical asymptotes in the graph of \( \sec x \). Here, the function's value can infinitely increase or decrease, approaching either \( +\infty \) or \( -\infty \), thereby confirming the presence of vertical asymptotes. It's crucial to comprehend how the function behaves near these points to accurately understand and depict its graph.

Identifying Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) tends towards \( \infty \) or \( -\infty \). They represent the value that the graph of the function approaches. For \( g(x) = \sec x \), the periodic nature of \( \sec x \) means that as \( x \) extends towards positive or negative infinity, it repeatedly encounters vertical asymptotes, barring any tendency towards a specific horizontal line. Consequently, \( \sec x \) does not have horizontal asymptotes. This is an essential insight for graphing periodic functions, indicating that their long-term behavior is not dominated by the kind of end behavior common in non-periodic rational functions.

Exploring Slant Asymptotes

Slant asymptotes, or oblique asymptotes, happen when a rational function's degree of the numerator is exactly one more than that of its denominator, allowing the function's graph to approach a slanted line as \( x \) tends towards either infinity. In the case of \( g(x) = \sec x \), which is periodic rather than polynomial or rational in a classical sense, slant asymptotes do not apply. The definition of slant asymptotes involves polynomials, whereas \( \sec x \)'s periodic behavior ensures it consistently returns to its starting pattern, rather than veering off to align along a single slant. Therefore, like horizontal asymptotes, slant asymptotes are not applicable to the function \( g(x) = \sec x \). Understanding this exception is key when analyzing non-polynomial relationships and their asymptotic behaviors.