Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 57

Chapter 4, Problem 57

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

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Identify the domain of the function by determining the values of \(x\) that make the denominator zero. For \(f(x) = \frac{4x}{x - 2}\), set the denominator equal to zero: \(x - 2 = 0\) and solve for \(x\).

Find the vertical asymptote(s) by using the values of \(x\) that are excluded from the domain. These occur where the denominator is zero, so the vertical asymptote is at \(x = 2\).

Determine the horizontal asymptote by comparing the degrees of the numerator and denominator. Since both numerator and denominator are degree 1, divide the leading coefficients to find the horizontal asymptote: \(y = \frac{4}{1} = 4\).

Calculate the \(x\)-intercept(s) by setting the numerator equal to zero and solving for \(x\). For \(4x = 0\), solve for \(x\) to find the \(x\)-intercept.

Calculate the \(y\)-intercept by evaluating \(f(0)\). Substitute \(x = 0\) into the function to find the corresponding \(y\)-value.

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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), where Q(x) ≠ 0. Understanding the form helps identify domain restrictions and behavior, such as vertical asymptotes where the denominator is zero.

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.

Graphing Steps for Rational Functions

Graphing involves identifying domain restrictions, intercepts, asymptotes, and plotting points to understand the function's shape. Following systematic steps ensures an accurate sketch of the function's behavior across its domain.

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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 + 1)/(x + 3) < 2

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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

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)=2x⁴−3x³−7x²−8x+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. (x + 4)/(2x - 1) ≤ 3

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Textbook Question

Use transformations of f(x) = (1/x) or f(x) = (1/x²) to graph each rational function. g(x) = 1/(x + 2)² - 1

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Textbook Question

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. h(x)=1/(x−3)²+1

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