College Algebra (10th Edition)

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

Chapter 5 - Section 5.2 - Properties of Rational Functions - 5.2 Assess Your Understanding - Page 351: 41

Answer

(a) 1st step (black graph): The parent function: $\dfrac{1}{x^2}$ 2nd step (green graph): Stretched by a factor of 2. $\dfrac{2}{x^2}$ 3rd step (red graph): Translated by 3 units to the right. $\dfrac{2}{(x-3)^2}$ Final step (blue graph): Translated 1 unit up. $1+\dfrac{2}{(x-3)^2}$ (b) The domain is $(-\infty,3)\cap(3,\infty)$. The range is $(1,\infty)$. (c) The only horizontal asymptote is y=1. The only vertical asymptote is x=3. 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 $(-\infty,3)\cap(3,\infty)$ The range is a vertical span from the function's smallest value of f(x) to the function's largest value of f(x). If there is a discontinuity, the range must show where the discontinuity happens. For example, if there is a horizontal asymptote on y=-4, the domain would be $(-\infty,-4)\cap(-4,\infty)$ The x-intercepts are all points of a graph when f(x)=0 while the y-intercepts are all points of a graph when x=0. Horizontal asymptotes are horizontal lines that approach a graph but never intersect it. Vertical asymptotes are vertical lines that approach a graph but never intersect it. Oblique asymptotes are diagonal lines that approach a graph and may intersect it.
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