Textbook Question
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+x−6)/(x−3)
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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 83
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2+1)/x
Verified step by step guidance1
Identify the given rational function: \(f(x) = \frac{x^{2} + 1}{x}\).
To find the slant (oblique) asymptote, perform polynomial long division of the numerator by the denominator: divide \(x^{2} + 1\) by \(x\).
Set up the division: \(x\) divides into \(x^{2}\) exactly \(x\) times. Multiply \(x\) by \(x\) to get \(x^{2}\), subtract this from \(x^{2} + 1\) to find the remainder.
The remainder after subtracting is \(1\). So, the division gives \(x\) with a remainder of \(1\), which can be written as \(f(x) = x + \frac{1}{x}\).
The slant asymptote is the quotient without the remainder term, so it is the line \(y = x\). This line describes the behavior of \(f(x)\) as \(x\) approaches infinity or negative infinity.
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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 behavior of rational functions, including their domains and asymptotes, is essential for graphing and analyzing their properties.
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Intro to Rational Functions
Slant (Oblique) Asymptotes
Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. They represent the line that the graph approaches as x approaches infinity or negative infinity, found by performing polynomial division.
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6:24Introduction to Asymptotes
Polynomial Division
Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. It helps find the quotient and remainder, which are used to determine slant asymptotes and simplify rational functions for graphing.
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Guided course
05:13Introduction to Polynomials
Related Practice
Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Let f(x)=an(x4−3x2−4). If f(3)=−150, determine the value of a_n.
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Textbook Question
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x
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Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x+6=0
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Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. x2 + 3x - 10 > 0
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Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0
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