Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $x=-x$.
$\begin{align}
& f\left( x \right)=\frac{{{x}^{2}}+x-12}{{{x}^{2}}-4} \\
& f\left( -x \right)=\frac{{{\left( -x \right)}^{2}}+\left( -x \right)-12}{{{\left( -x \right)}^{2}}-4} \\
& =\frac{{{x}^{2}}-x-12}{{{x}^{2}}-4}
\end{align}$
Therefore, the function $f\left( -x \right)$ is not equal to $-f\left( x \right)$ and the $f\left( x \right)$. So, the graph of the function is not symmetrical about the $y$ axis and the origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$.
$\begin{align}
& \frac{{{x}^{2}}+x-12}{{{x}^{2}}-4}=0 \\
& x=3,-4
\end{align}$ ,
Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$.
$f\left( 0 \right)=3$
Step 4: Since the degree of the numerator is equal to the denominator, the horizontal asymptote is:
$y=1$
Step 5: For the vertical asymptote, equate the denominator to 0.
$x=\pm 2$
