Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Question

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

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} .

Chapter 3.7, Problem 54E

Chapter 3.7, Problem 56E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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