Problem 58 - Algebra and Trigonometry

Question

58–59 ? Graph the function and comment on vertical and horizontal asymptotes. $$y=2^{1 / x}$$

Step-by-Step Solution

Answer

Vertical asymptote: \( x = 0 \); Horizontal asymptote: \( y = 1 \).

Step 1: Understand the Function

The function we are given is \( y = 2^{1/x} \). This is an exponential function where the base is 2 and the exponent is the reciprocal of \( x \). Our task involves analyzing the behavior of this function, specifically focusing on any asymptotes it might have.

Step 2: Identify the Horizontal Asymptote

For horizontal asymptotes, we analyze the behavior of \( y = 2^{1/x} \) as \( x \) approaches positive or negative infinity. As \( x \to \infty \), \( 1/x \to 0 \) and thus \( y = 2^{1/x} \to 2^0 = 1 \). Hence, there is a horizontal asymptote at \( y = 1 \). As \( x \to -\infty \), \( 1/x \to 0 \), again \( y = 1 \), reinforcing the same horizontal asymptote.

Step 3: Identify the Vertical Asymptote

For vertical asymptotes, we look for values of \( x \) where the function is undefined or behaves erratically. The function \( y = 2^{1/x} \) does not have a defined output at \( x = 0 \), as \( 1/0 \) is undefined. Thus, \( x = 0 \) is a vertical asymptote because as \( x \to 0^+ \) or \( x \to 0^- \), the exponent \( 1/x \) approaches \( +\infty \) and \( -\infty \), respectively, causing \( y \) to shoot up towards infinity or drop down towards zero.

Step 4: Sketch the Graph

To sketch \( y = 2^{1/x} \), start by plotting key points and asymptotes. The horizontal asymptote is the line \( y = 1 \). The vertical asymptote is the line \( x = 0 \). Choose points like \( x = 1 \), \( 2 \), \( -1 \), and \( -2 \), and compute \( y \). For example, when \( x = 1 \), \( y = 2^1 = 2 \); when \( x = 2 \), \( y = 2^{0.5} = \sqrt{2} \); when \( x = -1 \), \( y = 2^{-1} = 0.5 \); when \( x = -2 \), \( y = 2^{-0.5} \approx 0.707 \). This provides points to help visualize the function's curve approaching the asymptotes.

Step 5: Comment on the Asymptotes

The function \( y = 2^{1/x} \) has one vertical asymptote at \( x = 0 \) because the exponent approaches values that make the function undefined. It also has a horizontal asymptote at \( y = 1 \) because as \( x \) approaches both \( +\infty \) and \( -\infty \), the value of \( y \) approaches 1.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values, or \( x \), approach positive or negative infinity. For the function \( y = 2^{1/x} \), we must consider what happens to \( y \) when \( x \) grows larger in both the positive and negative directions. - As \( x \to \infty \), the value of \( 1/x \) approaches 0 since dividing by a very large number makes it increasingly smaller. Consequently, \( y = 2^{1/x} \) also approaches \( 2^0 = 1 \). - Similarly, if \( x \to -\infty \), \( 1/x \) again tends towards 0 despite the negative direction, leading \( y \) to approach 1 once more.Thus, we identify a horizontal asymptote at \( y = 1 \). Regardless of whether \( x \) is heading infinitely positive or negative, the output of the function comes closer to 1.

Vertical Asymptotes

Vertical asymptotes are about the values of \( x \) where a function becomes undefined or exhibits extreme behavior, such as shooting off to infinity. In the function \( y = 2^{1/x} \), these considerations focus on the value \( x = 0 \).- At \( x = 0 \), the term \( 1/x \) becomes undefined, resulting in the whole expression \( 2^{1/x} \) lacking a defined numerical output.- As \( x \to 0^+ \) (from the right), \( 1/x \to +\infty \), pushing \( y \) towards extremely high values.- Conversely, as \( x \to 0^- \) (from the left), \( 1/x \to -\infty \), forcing \( y \) towards zero.Thus, the vertical asymptote occurs at \( x = 0 \), dividing the graph into distinct behaviors on either side, where the function either explodes towards high values or near-zero levels, underscoring the boundary where the function's behavior sharply changes.

Graphing Functions

When graphing \( y = 2^{1/x} \), understanding both horizontal and vertical asymptotes is crucial to shaping the graph's behavior. To graph this function correctly:- **Horizontal Asymptote**: Plot the horizontal line at \( y = 1 \) to represent where the function levels off at both extremes of \( x \).- **Vertical Asymptote**: Mark the line \( x = 0 \), showing where the function is undefined, creating a visual division in the graph.To sketch the curve, calculate specific function values:

• For \( x = 1 \), \( y = 2^1 = 2 \), indicating a point above the horizontal asymptote.
• For \( x = 2 \), \( y = \sqrt{2} \approx 1.414 \), closer to the asymptote, dropping as \( x \) increases.
• For \( x = -1 \), \( y = 0.5 \), showing the curve under the asymptote.
• For \( x = -2 \), \( y \approx 0.707 \), moving nearer to 1 as \( x \) moves further negative.

These points assist in drawing the curve, revealing that as \( x \) moves away from 0 in either direction, the function aligns closer to its horizontal asymptote, while distinctly avoiding the vertical asymptote line. This understanding integrates the asymptotic lines into the graph, aiding clear visualization of the function's behavior.