Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.6 - Rational Functions and Their Graphs - Exercise Set - Page 399: 71

Answer

Graph of the function is:
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Work Step by Step

Step 1: Substitute $x=-x$ $\begin{align} & f\left( x \right)=\frac{x+2}{{{x}^{2}}+x-6} \\ & f\left( -x \right)=\frac{\left( -x \right)+2}{{{\left( -x \right)}^{2}}+\left( -x \right)-6} \\ & =\frac{-x+2}{{{x}^{2}}-x-6} \end{align}$ Therefore, the function $f\left( -x \right)$ is not equal to either $-f\left( x \right)$ or $f\left( x \right)$. Hence, the graph of the function is symmetrical neither about the $y$ -axis nor about the origin. Step 2: To calculate the x intercepts equate $f\left( x \right)=0$. $\begin{align} & \frac{x+2}{{{x}^{2}}+x-6}=0 \\ & x=-2 \end{align}$ , Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$. $\begin{align} & f\left( 0 \right)=\frac{2}{-6} \\ & f\left( 0 \right)=\frac{-1}{3} \\ \end{align}$ Step 4: Since the degree of the numerator is less than the denominator, there is no horizontal asymptote. Step 5: For the vertical asymptote, equate the denominator to 0. $x=2,-3$
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