## Precalculus (6th Edition) Blitzer

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}