Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Chapter 3.7, Problem 55E
To determine

To find: the intercepts and asymptotes, and then sketch a graph of the rational function. To state the domain and range.

Expert Solution & Answer
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Answer to Problem 55E

The y intercept is 1.

The vertical asymptote occurs at x=−1 .

The domain of the function is equal to {x|x≠−1} .

The range of the function is equal to {y|y≥0} .

Explanation of Solution

Given information:

Given function,

  r(x)=x2−2x+1x2+2x+1

Calculation:

Consider the function,

  r(x)=x2−2x+1x2+2x+1

To solve for the y intercept, simplify need to plug zero in for x .

Consider,

  x2−2x+1x2+2x+1=(0)2−2(0)+1(0)+2(0)+1=1

Therefore, the y intercept is 1.

To solve for the x intercept, set the function equal to 0 and solve for x .

Consider,

  x2−2x+1x2+2x+1=0x2−2x+1=0(x−1)2=0x=1

Vertical asymptotes:

The vertical asymptote occurs at the point that makes the denominator of the function zero.

The denominator of the function is,

  x2+2x+1

Find the vertical asymptotes,

  x2+2x+1=0(x+1)2=0x=−1

Therefore, the vertical asymptote occurs at x=−1 .

Horizontal asymptotes:

The degree of the numerator and denominator are the same, the horizontal asymptote occurs at the point determined by the following equation:

  Leading Coefficient of NumeratorLeading Coefficient of Denominator=11=1

The horizontal asymptote takes place at 1.

The equation for the asymptote is y=1 .

Sketch the graph of the function r(x)=x2−2x+1x2+2x+1 .

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 55E

From the graph,

The vertical asymptote at x=−1 .

The domain of the function is equal to {x|x≠−1} .

The horizontal asymptote at y=1 , from the graph,

The range of the function is equal to {y|y≥0} .

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Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 3.1 - To put the quadratic function f(x)=ax2+bx+c in...Ch. 3.1 - The quadratic function f(x) = a(x - h)2 + k is in...Ch. 3.1 - The graph of f(x) = 3(x - 2)2 - 6 is a parabola...Ch. 3.1 - The graph of f(x) = -3(x - 2)2 - 6 is a parabola...Ch. 3.1 - Prob. 5ECh. 3.1 - Prob. 6ECh. 3.1 - Prob. 7ECh. 3.1 - Prob. 8ECh. 3.1 - Prob. 9ECh. 3.1 - Prob. 10E
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Prob. 43ECh. 3.7 - Prob. 44ECh. 3.7 - Prob. 45ECh. 3.7 - Prob. 46ECh. 3.7 - Prob. 47ECh. 3.7 - Prob. 48ECh. 3.7 - Prob. 49ECh. 3.7 - Prob. 50ECh. 3.7 - Prob. 51ECh. 3.7 - Prob. 52ECh. 3.7 - Prob. 53ECh. 3.7 - Prob. 54ECh. 3.7 - Prob. 55ECh. 3.7 - Prob. 56ECh. 3.7 - Prob. 57ECh. 3.7 - Prob. 58ECh. 3.7 - Prob. 59ECh. 3.7 - Prob. 60ECh. 3.7 - Prob. 61ECh. 3.7 - Prob. 62ECh. 3.7 - Prob. 63ECh. 3.7 - Prob. 64ECh. 3.7 - Prob. 65ECh. 3.7 - Prob. 66ECh. 3.7 - Prob. 67ECh. 3.7 - Prob. 68ECh. 3.7 - Prob. 69ECh. 3.7 - Prob. 70ECh. 3.7 - Prob. 71ECh. 3.7 - Prob. 72ECh. 3.7 - Prob. 73ECh. 3.7 - Prob. 74ECh. 3.7 - Prob. 75ECh. 3.7 - Prob. 76ECh. 3.7 - Prob. 77ECh. 3.7 - Prob. 78ECh. 3.7 - Prob. 79ECh. 3.7 - Prob. 80ECh. 3.7 - Prob. 81ECh. 3.7 - Prob. 82ECh. 3.7 - Prob. 83ECh. 3.7 - Drug Concentration After a certain drug is...Ch. 3.7 - Drug Concentration A drug is administered to a...Ch. 3.7 - Flight of a Rocket Suppose a rocket is fired...Ch. 3.7 - The Doppler Effect As a train moves toward an...Ch. 3.7 - Prob. 88ECh. 3.7 - Prob. 89ECh. 3.7 - Prob. 90ECh. 3.7 - Prob. 91ECh. 3.7 - Prob. 92ECh. 3 - Prob. 1RCCCh. 3 - Prob. 2RCCCh. 3 - Prob. 3RCCCh. 3 - Prob. 4RCCCh. 3 - Prob. 5RCCCh. 3 - Prob. 6RCCCh. 3 - Prob. 7RCCCh. 3 - Prob. 8RCCCh. 3 - Prob. 9RCCCh. 3 - Prob. 10RCCCh. 3 - Prob. 11RCCCh. 3 - Prob. 12RCCCh. 3 - Prob. 13RCCCh. 3 - Prob. 1RECh. 3 - Prob. 2RECh. 3 - Prob. 3RECh. 3 - Prob. 4RECh. 3 - Prob. 5RECh. 3 - Prob. 6RECh. 3 - Prob. 7RECh. 3 - Prob. 8RECh. 3 - Prob. 9RECh. 3 - Prob. 10RECh. 3 - Prob. 11RECh. 3 - Prob. 12RECh. 3 - Prob. 13RECh. 3 - Prob. 14RECh. 3 - Prob. 15RECh. 3 - Prob. 16RECh. 3 - Prob. 17RECh. 3 - Prob. 18RECh. 3 - Prob. 19RECh. 3 - Prob. 20RECh. 3 - Prob. 21RECh. 3 - Prob. 22RECh. 3 - Prob. 23RECh. 3 - Prob. 24RECh. 3 - Prob. 25RECh. 3 - Prob. 26RECh. 3 - Prob. 27RECh. 3 - Prob. 28RECh. 3 - Prob. 29RECh. 3 - Prob. 30RECh. 3 - Prob. 31RECh. 3 - Prob. 32RECh. 3 - Prob. 33RECh. 3 - Prob. 34RECh. 3 - Prob. 35RECh. 3 - Prob. 36RECh. 3 - Prob. 37RECh. 3 - Prob. 38RECh. 3 - 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Graph the polynomial P(x) = (x + 2)3 + 27, showing...Ch. 3 - (a) Use synthetic division to find the quotient...Ch. 3 - Let P(x) = 2x3 5x2 4x + 3. 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Asymptotes - What are they? : ExamSolutions Maths Revision; Author: ExamSolutions;https://www.youtube.com/watch?v=5Hl_WJXcR6M;License: Standard YouTube License, CC-BY