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 61E

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 61E

y intercept is −2 .

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

The equation for the horizontal asymptote is y=3 .

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

The range of the function is equal to {y|y≤−1.5,y≥2.4} .

Explanation of Solution

Given information:

Given function,

r(x)=3x2+6x2−2x−3

Calculation:

Consider the function,

r(x)=3x2+6x2−2x−3

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

Consider,

3x2+6x2−2x−3=3(0)2+6(0)2−2(0)−3=−2

Therefore, y intercept is −2 .

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

Consider,

3x2+6x2−2x−3=03x2+6=0x=±2i

The function has no x -intercepts.

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−6=0(x−3)(x+1)=0x=−1,x=3

Therefore, the vertical asymptote occurs at x=−1,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=31=3

The equation for the asymptote is y=3 .

Sketch the graph of the function r(x)=3x2+6x2−2x−3 .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 61E

From the graph,

The vertical asymptote at x=−1,x=3 .

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

The horizontal asymptote at y=3 , from the graph,

The range of the function is equal to {y|y≤−1.5,y≥2.4} .

Chapter 3.7, Problem 60E

Chapter 3.7, Problem 62E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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