/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 In Exercises \(37-40\), use a gr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 38

In Exercises \(37-40\), use a graphing utility to graph the function and identify any horizontal asymptotes. $$ f(x)=\frac{|3 x+2|}{x-2} $$

Short Answer

Expert verified
The function \(f(x)=\frac{|3x+2|}{x-2}\) has two horizontal asymptotes, \(y=3\) and \(y=-3\), and a vertical asymptote at \(x=2\).

Step by step solution

01

Deal with absolute value

The absolute value can be defined as \(|x| = x\) for \(x \geq 0\) and \(|x| = -x\) for \(x < 0\). Thus the function can be divided into two parts: \(f_1(x) = \frac{3x+2}{x-2}\) for \(3x+2 \geq 0\) and \(f_2(x) = \frac{-3x-2}{x-2}\) for \(3x+2 < 0\).
02

Find asymptotes

To find the horizontal asymptotes of the function, we examine the limit of the function as \(x \to \infty\) and \(x \to -\infty\). For \(f_1(x) = \frac{3x+2}{x-2}\), as \(x \to \infty\), \(f_1(x) \to 3\), and as \(x \to -\infty\), \(f_1(x) \to 3\). So \(y=3\) is a horizontal asymptote. As for \(f_2(x) = \frac{-3x-2}{x-2}\), as \(x \to \infty\), \(f_2(x) \to -3\), and as \(x \to -\infty\), \(f_2(x) \to -3\). So \(y=-3\) is a horizontal asymptote. Note that a vertical asymptote happens where the denominator of our function equals zero. So for \(x=2\) there's a vertical asymptote.
03

Graph the function

Now, plot the function \(f_1(x) = \frac{3x+2}{x-2}\) for \(3x+2 \geq 0\) and \(f_2(x) = \frac{-3x-2}{x-2}\) for \(3x+2 < 0\) to double check the calculated values. A graphing utility such as a graphic calculator or a computer program like GeoGebra can be used for this task.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Functions
An absolute value function involves expressions like \(|x|\), which measure the distance of a number from zero on the number line. It's always positive or zero, making it quite different from regular functions that can have negative outputs. Absolute value can be determined by:
  • \(|x| = x\) for \(x \geq 0\)
  • \(|x| = -x\) for \(x < 0\)
In the function \(f(x) = \frac{|3x+2|}{x-2}\), the absolute value makes the function behave differently depending on the sign of \(3x + 2\). This requires us to split the function into two parts: one for when \(3x + 2 \geq 0\) and another for when \(3x + 2 < 0\). Understanding the behavior of absolute value functions is crucial to analyze the full range of the given function.
Limits at Infinity
Limits at infinity are used to understand the behavior of a function as the input value grows very large positively or negatively. For rational functions like \(f(x) = \frac{3x+2}{x-2}\), we focus on the highest power of \(x\) in the numerator and denominator to determine these limits.
  • As \(x \to \infty\), \(f(x) \sim \frac{3x}{x} = 3\)
  • As \(x \to -\infty\), \(f(x) \sim \frac{-3x}{x} = -3\)
These simplified forms indicate that the behavior at these extremes remains constant, indicating the existence of horizontal asymptotes. Identifying these limits helps determine where the function levels out, which are important for graphing and understanding the long-term behavior of functions.
Vertical Asymptotes
Vertical asymptotes occur where the denominator of a function equals zero, causing the function to become undefined, often leading to a graph rising or falling sharply. For our function \(f(x) = \frac{|3x+2|}{x-2}\), the denominator \(x-2\) equals zero when \(x=2\). When this happens, neither \(f_1(x)\) nor \(f_2(x)\) is defined because you can't divide by zero. Thus, \(x=2\) is a vertical asymptote, and the graph will show a sharp discontinuity at this point, meaning the function rises or falls towards infinity as it approaches \(x=2\) from either side. Recognizing vertical asymptotes is vital for sketching the accurate graph of a function.
Graphing Rational Functions
Graphing rational functions like \(f(x) = \frac{|3x+2|}{x-2}\) requires an understanding of both horizontal and vertical asymptotes, along with considering absolute values that influence the graph's behavior. When graphing, it鈥檚 essential to:
  • Identify any asymptotes: both horizontal \(y=3\) and \(y=-3\), and the vertical \(x=2\).
  • Determine the intervals for absolute values and split the function into parts.
  • Use a graphing utility to accurately draw the curve and visualize behavior near asymptotes.
Graphing utilities can simplify this job by allowing you to input complex functions and view their graphs on a screen. This can confirm analytical predictions and provide a precise visual understanding for often complex behaviors of rational functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{1}{x^{2}-x-2} $$

Use the definitions of increasing and decreasing functions to prove that \(f(x)=x^{3}\) is increasing on \((-\infty, \infty)\).

A model for the specific gravity of water \(S\) is \(S=\frac{5.755}{10^{8}} T^{3}-\frac{8.521}{10^{6}} T^{2}+\frac{6.540}{10^{5}} T+0.99987,0

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The maximum slope of the graph of \(y=\sin (b x)\) is \(b\).

Assume that \(f\) is differentiable for all \(x\). The signs of \(f^{\prime}\) are as follows. \(f^{\prime}(x)>0\) on \((-\infty,-4)\) \(f^{\prime}(x)<0\) on (-4,6) \(f^{\prime}(x)>0\) on \((6, \infty)\) Supply the appropriate inequality for the indicated value of \(c\). $$ g(x)=-f(x) \quad g^{\prime}(-6) \quad 0 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.