Answer
Graph the function is as:
Work Step by Step
Step 1: Substitute $x=-x$.
$\begin{align}
& f\left( x \right)=\frac{2{{x}^{2}}}{{{x}^{2}}-1} \\
& f\left( -x \right)=\frac{2{{\left( -x \right)}^{2}}}{{{\left( -x \right)}^{2}}-1} \\
& =\frac{2{{x}^{2}}}{{{x}^{2}}-1} \\
& =f\left( x \right)
\end{align}$
Therefore, the function $f\left( -x \right)$ is equal to $f\left( x \right)$. So, the graph of the function is either symmetrical about the $y$ axis or origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$\begin{align}
& \frac{2{{x}^{2}}}{{{x}^{2}}-1}=0 \\
& x=0
\end{align}$
Step 3: To calculate the y intercepts equate $f\left( 0 \right)$
$\begin{align}
& f\left( 0 \right)=\frac{2\left( 0 \right)}{\left( 0 \right)-1} \\
& f\left( 0 \right)=0 \\
\end{align}$
Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is $y=2$.
Step 5: For the vertical asymptote, equate the denominator to 0.
$\begin{align}
& x-1=0 \\
& x=1
\end{align}$
