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 56E

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 56E

The y intercept is 0.

The vertical asymptote occurs at x=3,x=−1 .

The equation for the asymptote is y=4 .

The domain of the function is equal to {x|x≠−1,x≠3} .

The range of the function is equal to {y|0<y,y≥2.9} .

Explanation of Solution

Given information:

Given function,

r(x)=4x2x2−2x−3

Calculation:

Consider the function,

r(x)=4x2x2−2x−3

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

Consider,

4x2x2−2x−3=4(0)2(0)2−2(0)−3=0

Therefore, the y intercept is 0.

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

Consider,

4x2x2−2x−3=04x2=0x=0

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−3

Find the vertical asymptotes,

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

Therefore, the vertical asymptote occurs at x=3,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=41=4

The equation for the asymptote is y=4 .

Sketch the graph of the function r(x)=4x2x2−2x−3 .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 56E

From the graph,

The vertical asymptote at x=−1 , and x=3 .

The domain of the function is equal to {x|x≠−1,x≠3} .

The horizontal asymptote at y=4 , from the graph,

The range of the function is equal to {y|0<y,y≥2.9} .

Chapter 3.7, Problem 55E

Chapter 3.7, Problem 57E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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