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 57E

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 57E

The y intercept is 2.

The vertical asymptote occurs at x=−3,x=2 .

The equation for the asymptote is y=2 .

The domain of the function is equal to {x|x≠−3,x≠2} .

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

Explanation of Solution

Given information:

Given function,

r(x)=2x2+10x−12x2+x−6

Calculation:

Consider the function,

r(x)=2x2+10x−12x2+x−6

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

Consider,

2x2+10x−12x2+x−6=2(0)2+10(0)−12(0)2+(0)−6=2

Therefore, the y intercept is 2.

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

Consider,

2x2+10x−12x2+x−6=02x2+10x−12=0x2+5x−6=0x2+6x−x−6=0x=−6,x=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+x−6

Find the vertical asymptotes,

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

Therefore, the vertical asymptote occurs at x=−3,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=21=2

The equation for the asymptote is y=2 .

Sketch the graph of the function r(x)=2x2+10x−12x2+x−6 .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 57E

From the graph,

The vertical asymptote at x=−3,x=2 .

The domain of the function is equal to {x|x≠−3,x≠2} .

The horizontal asymptote at y=2 , from the graph,

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

Chapter 3.7, Problem 56E

Chapter 3.7, Problem 58E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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