College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter 3, Polynomial and Rational Functions - Section 3.6 - Rational Functions - 3.6 Exercises - Page 344: 15

Answer

The vertical asymptote moves to $x=-1$. The horizontal asymptote remains $y=0.$ The domain changes to $(-\infty,-1)\cup(-1,\infty)$ The range remains $(-\infty,0)\cup(0,\infty)$

Work Step by Step

Let $f(x)=\displaystyle \frac{1}{x}$ The asymptotes are $x=0$ (vertical) and $y=0$ (horizontal). The domain and range are both $\mathbb{R}/\{0\}=(-\infty,0)\cup(0,\infty)$. The graph and the asymptotes of f are graphed with red dashed lines. $r(x)=\displaystyle \frac{3}{x+1}=3\cdot f(x+1),$ so its graph (blue solid line) is obtained from the graph of f by -shifting it to the left by $1$ unit, to obtain $f(x+1)$ and then, - vertically stretching by a factor of 3. The vertical asymptote moves to $x=-1$. The horizontal asymptote remains $y=0.$ The domain changes to $(-\infty,-1)\cup(-1,\infty)$ The range remains $(-\infty,0)\cup(0,\infty)$
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