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 43

Chapter 4, Problem 43

Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)

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Identify the degrees of the numerator and denominator polynomials in the rational function \(f(x) = \frac{-2x + 1}{3x + 5}\). Here, both the numerator and denominator are first-degree polynomials (degree 1).

Recall the rule for horizontal asymptotes of rational functions: - If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). - If the degrees are equal, the horizontal asymptote is \(y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\). - If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique asymptote instead).

Since the degrees of numerator and denominator are equal (both 1), find the leading coefficients: numerator's leading coefficient is \(-2\), denominator's leading coefficient is \(3\).

Write the horizontal asymptote as \(y = \frac{-2}{3}\) by dividing the leading coefficients.

Conclude that the horizontal asymptote of the function \(f(x) = \frac{-2x + 1}{3x + 5}\) is the horizontal line \(y = \frac{-2}{3}\).

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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 the behavior of rational functions, especially their graphs, is essential for analyzing asymptotes and limits.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a function as x approaches infinity or negative infinity. It is a horizontal line y = L that the graph approaches but does not necessarily touch, indicating the end behavior of the function.

Degree of Polynomials in Numerator and Denominator

The degrees of the numerator and denominator polynomials determine the horizontal asymptote of a rational function. If degrees are equal, the asymptote is the ratio of leading coefficients; if numerator degree is less, asymptote is y=0; if greater, no horizontal asymptote.

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Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x⁴−2x³+x²+12x+8

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Textbook Question

Solve the equation 12x³+16x²−5x−3=0 given that -3/2 is a root.

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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+3)/(x+4)<0

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Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=5x²−5x

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Textbook Question

Solve the equation 2x³−3x²−11x+6=0 given that -2 is a zero of f(x)=2x³−3x²−11x+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)/(x+3) > 0

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