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: 59

Answer

The graph of the function is:
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Work Step by Step

Step 1: Substitute $x=-x$ $\begin{align} & f\left( x \right)=\frac{2x}{{{x}^{2}}-4} \\ & f\left( -x \right)=\frac{2\left( -x \right)}{{{\left( -x \right)}^{2}}-4} \\ & =\frac{-2x}{{{x}^{2}}-4} \\ & =-f\left( x \right) \end{align}$ Therefore, the function $f\left( -x \right)$ is equal to $-f\left( x \right)$. So, the graph of the function is symmetrical about the $y$ axis and origin. Step 2: To calculate the x intercepts, equate $f\left( x \right)=0$. $\begin{align} & \frac{2x}{{{x}^{2}}-4}=0 \\ & x=0 \end{align}$ Step 3: To calculate the y intercepts, evaluate $f\left( 0 \right)$. $\begin{align} & f\left( 0 \right)=\frac{2\left( 0 \right)}{{{\left( 0 \right)}^{2}}-4} \\ & f\left( 0 \right)=0 \\ \end{align}$ 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. $\begin{align} & {{x}^{2}}-4=0 \\ & x=\pm 2 \end{align}$
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