Textbook Question
Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 3/(x +3) > 3/(x - 2)
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 65
Follow the seven steps to graph each rational function. f(x)=− 1/(x2−4)
Verified step by step guidance1
Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Set the denominator equal to zero: \(x^{2} - 4 = 0\).
Solve the equation \(x^{2} - 4 = 0\) by factoring it as \((x - 2)(x + 2) = 0\), which gives the values \(x = 2\) and \(x = -2\). These values are excluded from the domain because they make the denominator zero, causing vertical asymptotes.
Determine the vertical asymptotes by noting that the function is undefined at \(x = 2\) and \(x = -2\). So, draw vertical dashed lines at these \(x\)-values.
Find the horizontal asymptote by analyzing the degrees of the numerator and denominator. The numerator is a constant (\(-1\)), and the denominator is a quadratic (\(x^{2} - 4\)). Since the degree of the denominator is greater than the numerator, the horizontal asymptote is \(y = 0\).
Create a table of values by choosing \(x\)-values around the vertical asymptotes (for example, values less than \(-2\), between \(-2\) and \(2\), and greater than \(2\)), then calculate the corresponding \(f(x)\) values to understand the behavior of the graph in each interval.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, zeros, and behavior is essential, especially where the denominator equals zero, causing vertical asymptotes or undefined points.
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Intro to Rational Functions
Asymptotes of Rational Functions
Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.
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6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic approach ensures an accurate representation of the function.
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8:19How to Graph Rational Functions
Related Practice
Textbook Question
Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. |x2 + 2x - 36| > 12
Textbook Question
Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?
Textbook Question
Follow the seven steps to graph each rational function. f(x)=2/(x2+x−2)
Textbook Question
In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. g(x) = (4x^2 - 16x + 16)/(2x - 3)
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Textbook Question
Find the domain of each function.
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