Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5
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 77
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x2+x−12)/(x2−4)
Verified step by step guidance1
Identify the domain of the function by finding the values of that make the denominator zero. Solve to find these values, since division by zero is undefined.
Factor both the numerator and denominator to simplify the function if possible. Factor and to their binomial factors.
Determine the vertical asymptotes by setting the denominator equal to zero and solving for . These are the values excluded from the domain where the function may approach infinity or negative infinity.
Find the horizontal or oblique asymptote by comparing the degrees of the numerator and denominator polynomials. Use the degree rules to determine the end behavior of the function.
Calculate the x-intercepts by setting the numerator equal to zero and solving for , and find the y-intercept by evaluating if it is in the domain.
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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 its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values and to analyze the function's behavior.
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Intro to Rational Functions
Finding Asymptotes
Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero (and numerator is nonzero), while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.
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6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing involves seven steps: determining domain, intercepts, asymptotes, sign analysis, and plotting points. This systematic approach helps visualize the function's shape and behavior accurately.
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Related Practice
Textbook Question
Use synthetic division to show that 5 is a solution of x^4−4x^3−9x^2+16x+20=0. Then solve the polynomial equation.
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Textbook Question
Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?
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Textbook Question
Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?
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Textbook Question
Follow the seven steps to graph each rational function. f(x)=x4/(x2+2)
Textbook Question
When 2x2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.
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