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 62E

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 62E

y intercept is 54 .

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

The equation for the asymptote is y=5 .

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

The range of the function is equal to {y|y≥1} .

Explanation of Solution

Given information:

Given function,

r(x)=5x2+5x2+4x+4

Calculation:

Consider the function,

r(x)=5x2+5x2+4x+4

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

Consider,

5x2+5x2+4x+4=5(0)2+5(0)2+4(0)+4=54

Therefore, y intercept is 54 .

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

Consider,

5x2+5x2+4x+4=05x2+5=0x=±i

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+4x+4

Find the vertical asymptotes,

x2+4x+4=0(x+2)(x+2)=0x=−2,x=−2

Therefore, the vertical asymptote occurs at x=−2,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=51=5

The equation for the asymptote is y=5 .

Sketch the graph of the function r(x)=5x2+5x2+4x+4 .

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.7, Problem 62E

From the graph,

The vertical asymptote at x=−2,x=−2 .

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

The horizontal asymptote at y=5 , from the graph,

The range of the function is equal to {y|y≥1} .

Chapter 3.7, Problem 61E

Chapter 3.7, Problem 63E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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