Textbook Question
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
2x3 + 11x2 < 7x + 6
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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 71
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x+2)/(x2+x−6)
Verified step by step guidance1
Identify the domain of the function by finding the values of \( x \) that make the denominator zero. Solve \( x^2 + x - 6 = 0 \) by factoring or using the quadratic formula.
Find the vertical asymptotes by setting the denominator equal to zero and excluding any values that also make the numerator zero (which would indicate a hole instead).
Determine the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator polynomials. Since the numerator is degree 1 and the denominator is degree 2, the horizontal asymptote is \( y = 0 \).
Find the x-intercepts by setting the numerator equal to zero and solving for \( x \). For \( f(x) = \frac{x+2}{x^2 + x - 6} \), set \( x + 2 = 0 \) and solve.
Find the y-intercept by evaluating \( f(0) \), which is \( \frac{0 + 2}{0^2 + 0 - 6} \). This gives the point where the graph crosses the y-axis.
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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, where the denominator Q(x) ≠ 0, is essential because values that make Q(x) zero are excluded and often correspond to vertical asymptotes or holes in the graph.
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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 the numerator is nonzero, while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.
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6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes. Plotting key points and analyzing limits help sketch an accurate graph, following a systematic approach such as the seven-step method.
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8:19How to Graph Rational Functions
Related Practice
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 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) ≤ - 3/4(x - 2)
Textbook Question
In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
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Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. 2x^2 + 9x + 4 ≥ 0
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Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x−2)/(x2−4)