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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 59
Chapter 4, Problem 59

Follow the seven steps to graph each rational function. f(x)=2x/(x2−4)

Verified step by step guidance
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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.
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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 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.
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Introduction 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).
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How to Graph Rational Functions
Related Practice
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. x/(x + 2) ≥ 2

Textbook Question

Find the inverse of f(x) = x3 + 2

Textbook Question

Solve: x+x+5=5\(\sqrt{x}\) + \(\sqrt{x + 5}\) = 5

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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

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. r(x) = (x^2 + 4x + 3)/(x + 2)^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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

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