Textbook Question
Solve:
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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 59
Follow the seven steps to graph each rational function. f(x)=2x/(x2−4)
Verified step by step guidance1
Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the equation \(x^{2} - 4 = 0\) to find these values.
Determine the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). These are the values excluded from the domain where the function tends to infinity.
Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the numerator is degree 1 and the denominator is degree 2, the horizontal asymptote is \(y = 0\).
Calculate the \(x\)-intercepts by setting the numerator equal to zero and solving for \(x\). This gives the points where the graph crosses the \(x\)-axis.
Calculate the \(y\)-intercept by evaluating \(f(0)\), substituting \(x=0\) into the function to find where the graph crosses the \(y\)-axis.
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Domain of a Rational Function
The domain of a rational function includes all real numbers except where the denominator equals zero. For f(x) = 2x/(x²−4), set the denominator x²−4 = 0 to find excluded values, which are x = ±2. Understanding the domain helps identify vertical asymptotes and restrictions on the graph.
Recommended video:
6:04
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 numerator is nonzero), while horizontal asymptotes describe end behavior based on degrees of numerator and denominator. For f(x) = 2x/(x²−4), vertical asymptotes are at x = ±2, and the horizontal asymptote is y = 0.
Recommended video:
6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing rational functions involves seven steps: finding the domain, intercepts, asymptotes, analyzing end behavior, plotting points, and sketching the curve. Each step builds understanding of the function’s behavior, ensuring an accurate and complete graph of f(x) = 2x/(x²−4).
Recommended video:
8:19How to Graph Rational Functions
Related Practice
Textbook Question
Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x5+2x4−15x3−10x2+12x+8
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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. f(x) = 2x/(x^2 - 9)
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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) = (2x - 4)/(x + 3)
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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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)
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Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation.1/(x - 3) < 1
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