Textbook Question
Among all pairs of numbers whose difference is 24, find a pair whose product is as small as possible. What is the minimum product?
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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 63
Follow the seven steps to graph each rational function. f(x)=−x/(x+1)
Verified step by step guidance1
Identify the function given: \(f(x) = \frac{-x}{x+1}\). This is a rational function where the numerator is \(-x\) and the denominator is \(x+1\).
Find the domain by determining where the denominator is zero. Set \(x+1=0\) and solve for \(x\) to find any vertical asymptotes or restrictions.
Find the vertical asymptote(s) by noting the values of \(x\) that make the denominator zero (from the domain step). These are the lines where the function is undefined.
Find 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.
Find the intercepts: For the \(y\)-intercept, evaluate \(f(0)\). For the \(x\)-intercept, set the numerator equal to zero and solve for \(x\).
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 domain restrictions, where the denominator equals zero, is essential to avoid undefined values and to identify vertical asymptotes.
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 horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.
Recommended video:
6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing involves identifying domain restrictions, intercepts, asymptotes, and plotting points to understand the function's behavior. Following a systematic seven-step process ensures a complete and accurate graph of the rational function.
Recommended video:
8:19How to Graph Rational Functions
Related Practice
Textbook Question
Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?
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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) = (4x^2 - 16x + 16)/(2x - 3)
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Textbook Question
Find the domain of each function.
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Textbook Question
Find the domain of each function.
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Textbook Question
Find the domain of each function.
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