Explanation Given information: The rational function is s ( x ) = 3 x 2 + 6 x 2 − 2 x − 3 . Calculation: The rational function s ( x ) can be written in its factor form as, s ( x ) = 3 ( x 2 + 2 ) ( x − 3 ) ( x + 1 ) ...... (1) The x- intercept of the rational function is the point such that the numerator of the function is zero. The numerator of the rational function s ( x ) is 3 x 2 + 6 . Here, the x - intercept of the rational function is greater than zero. Therefore, the x -intercept of the rational function does not exist. The y -intercept of the rational function are the points such that x is equals to zero. Substitute 0 for x in the equation (1) to obtain they -intercept of the rational function as, s ( 0 ) = 3 ( 0 2 + 2 ) ( 0 − 3 ) ( 0 + 1 ) = 6 − 3 = − 2 The value of s ( 0 ) is equal to − 2 . Therefore, the y -intercept of the rational function is − 2 . The vertical asymptote of the rational function is a line x = a such that y approaches to ± ∞ if x approaches to a. At the point of vertical asymptote the denominator of the rational function is zero. The denominator of the function s ( x ) from the equation (1) is ( x − 3 ) ( x + 1 ) . Now, equate the denominator of the rational function s ( x ) equals to zero to obtain the vertical asymptote of the rational function as, ( x − 3 ) ( x + 1 ) = 0 x − 3 = 0 or x + 1 = 0 x = 3 or x = − 1 Therefore, the vertical asymptote of the function is x = − 1 and x = 0 . The horizontal asymptote is the value of y such that x approaches to + ∞ or − ∞ . The horizontal asymptote of the rational function is y = b . Here, b is the ratio of coefficient of x that has highest degree in numerator and denominator of rational function s ( x ) as, y = ( 3 ) ⋅ ( 1 ) ( 1 ) ⋅ ( 1 ) y = 3 Therefore, the horizontal asymptote of the function is y = 3 . Graph: Obtain the value of y for different values of x to plot the graph of rational function s ( x ) . Substitute 0 for xin equation (1) to find the value of s ( 0 ) as, s ( 0 ) = 3 ( 0 2 + 2 ) ( 0 − 3 ) ( 0 + 1 ) = 6 − 3 = − 2 Substitute 1 for xin equation (1) to find the value of s ( 1 ) as follows, s ( 1 ) = 3 ( 1 2 + 2 ) ( 1 − 3 ) ( 1 + 1 ) = 9 − 4 = − 9 4 Substitute − 2 for xin equation (1) to find the value of s ( − 2 ) as follows, s ( − 2 ) = 3 ( ( − 2 ) 2 + 2 ) ( − 2 − 3 ) ( − 2 + 1 ) = 18 5 Similarly, obtain the other coordinate points that lie on the curve and list all the points in form of table as shown in Table 1

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 3.6, Problem 56E

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The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

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Chapter 3.6, Problem 55E

Chapter 3.6, Problem 57E

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The intercepts and asymptotes of the rational function s ( x ) . Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Solution Summary: The author explains the rational function s(x) and its intercepts and asymptotes.

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To find:

The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

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Chapter 3 Solutions

The intercepts and asymptotes of the rational function s ( x ) . Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Solution Summary: The author explains the rational function s(x) and its intercepts and asymptotes.

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To find: The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.

Want to see the full answer?

Chapter 3 Solutions

To find:

The intercepts and asymptotes of the rational function s(x). Sketch the graph of the function and also state its domain and range. Also, use a graphing device to confirm the answers.