College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 5 - Review Exercises - Page 397: 12

Answer

Domain: $(-\infty,-3)\cap(-3,3)\cap(3,\infty)$ There are two vertical asymptotes: x=-3 and x=3 There is only one horizontal asymptote: y=0. There are no oblique asymptotes.

Work Step by Step

The domain is a horizontal span from the function's smallest value of x to the function's largest value of x. If there is a discontinuity, the domain must show where the discontinuity happens. For example, if there is a vertical asymptote on x=3, the domain would be $(-\infty,3)\cap(3,\infty)$ To find asymptotes, first, we must make sure the function is in the lowest terms. We can see that the equation is in lowest terms. To find vertical asymptotes, we must find the values that make the denominator equal zero and solve for x. In this case: $x^2-9=0$ $x^2=9$ $\sqrt{x^2}=\sqrt9$ $x_1=-3$ $x_2=3$ There are two cases to determine horizontal asymptotes. If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote will be y = 0. If the degrees of both the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. In this case, we can see that the degree of the denominator is greater than the degree of the numerator, so the horizontal asymptote is y=0. To determine oblique asymptotes, the degree of the numerator must be one degree greater than the degree of the denominator. Since that requirement is not being met here, there are no oblique asymptotes.
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