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 67
Follow the seven steps to graph each rational function. f(x)=2/(x2+x−2)
Verified step by step guidance1
Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the quadratic equation \(x^{2} + x - 2 = 0\) to find these values, since the function is undefined there.
Factor the denominator \(x^{2} + x - 2\) into \((x + 2)(x - 1)\) to clearly see the zeros and potential vertical asymptotes at \(x = -2\) and \(x = 1\).
Determine the vertical asymptotes by setting each factor in the denominator equal to zero: \(x + 2 = 0\) and \(x - 1 = 0\), which gives \(x = -2\) and \(x = 1\) respectively.
Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the numerator is a constant (degree 0) and the denominator is degree 2, the horizontal asymptote is \(y = 0\).
Create a table of values by choosing \(x\)-values around the vertical asymptotes and plugging them into \(f(x) = \frac{2}{x^{2} + x - 2}\) to find corresponding \(y\)-values. This will help in sketching the graph accurately.
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16mKey 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 for graphing. The function is undefined where the denominator Q(x) equals zero, leading to vertical asymptotes or holes.
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6:04
Intro to Rational Functions
Factoring Quadratic Expressions
Factoring the quadratic denominator x² + x − 2 helps identify values that make the denominator zero. By rewriting it as (x + 2)(x − 1), we find critical points where the function is undefined, which correspond to vertical asymptotes or discontinuities in the graph.
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06:08Solving Quadratic Equations by Factoring
Asymptotes and Graph Behavior
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. Identifying these helps sketch the graph accurately and understand the function's limits.
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6:24Introduction to Asymptotes
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
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 5x - 3 < 0
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
Follow the seven steps to graph each rational function. f(x)=− 1/(x2−4)
Textbook Question
Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 1/(x + 1) > 2/(x - 1)
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