Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $x=-x$
$\begin{align}
& f\left( x \right)=\frac{-2}{{{x}^{2}}-1} \\
& f\left( -x \right)=\frac{-2}{{{\left( -x \right)}^{2}}-1} \\
& =\frac{-2}{{{x}^{2}}-1} \\
& =f\left( x \right)
\end{align}$
And,
$\begin{align}
& -f\left( x \right)=-\left( \frac{-2}{{{x}^{2}}-1} \right) \\
& =\frac{2}{{{x}^{2}}-1} \\
& \ne f\left( -x \right)
\end{align}$
Therefore, the graph of the function is symmetric about the $y$-axis but not about the origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$-\frac{2}{{{x}^{2}}-1}=0$ ,
Which is not true for any value of x. Thus, there are no x-intercepts.
Step 3: To calculate the y intercepts, evaluate $f\left( 0 \right)$.
$\begin{align}
& f\left( 0 \right)=\frac{-2}{\left( 0 \right)-1} \\
& f\left( 0 \right)=2 \\
\end{align}$
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.
$\begin{align}
& {{x}^{2}}-1=0 \\
& x=\pm 1
\end{align}$
