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 64E

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 64E

y intercept is 0.

The vertical asymptote occurs at x=−1,x=2 .

The equation for the horizontal asymptote is y=1 .

The domain of the function is equal to {x|x≠−1,x≠2} .

The range of the function is equal to {y|y∈ℝ} .

Explanation of Solution

Given information:

Given function,

t(x)=x3−x2x3−3x−2

Calculation:

Consider the function,

t(x)=x3−x2x3−3x−2

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

Consider,

x3−x2x3−3x−2=(0)3−(0)2(0)3−3(0)−2=0

Therefore, y intercept is 0.

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

Consider,

x3−x2x3−3x−2=0x3−x2=0x2(x−1)=0x=1,x=0

Vertical asymptotes:

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

The denominator of the function is,

x3−3x−2

Find the vertical asymptotes,

x3−3x−2=0x3+x2−x2−x−2x−2=0(x+1)2(x−2)=0x=2,x=−1

Therefore, the vertical asymptote occurs at x=−1,x=2 .

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 equation for the asymptote is y=1 .

Sketch the graph of the function t(x)=x3−x2x3−3x−2 .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 64E

From the graph,

The vertical asymptote at x=−1,x=2 .

The domain of the function is equal to {x|x≠−1,x≠2} .

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

The range of the function is equal to {y|y∈ℝ} .

Chapter 3.7, Problem 63E

Chapter 3.7, Problem 65E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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