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