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 59E

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 59E

The function has no y intercept.

The vertical asymptote occurs at x=0,x=−3 .

The equation for the horizontal asymptote is y=1 .

The domain of the function is equal to {x|x≠0,x≠−3} .

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

Explanation of Solution

Given information:

Given function,

r(x)=x2−x−6x2+3x

Calculation:

Consider the function,

r(x)=x2−x−6x2+3x

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

Consider,

x2−x−6x2+3x=(0)2−(0)−6(0)2+3(0)=∞

Therefore, the function has no y intercept.

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

Consider,

x2−x−6x2+3x=0x2−x−6=0x2−3x+2x−6=0(x−3)(x+2)=0x=−2,x=3

Vertical asymptotes:

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

The denominator of the function is,

x2+3x

Find the vertical asymptotes,

x2+3x=0x(x+3)=0x=0,x=−3

Therefore, the vertical asymptote occurs at x=0,x=−3 .

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 r(x)=x2−x−6x2+3x .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 59E

From the graph,

The vertical asymptote at x=0,x=−3 .

The domain of the function is equal to {x|x≠0,x≠−3} .

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

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

Chapter 3.7, Problem 58E

Chapter 3.7, Problem 60E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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