Answer
Graph the function as:
Work Step by Step
Step 1: Substitute $x=-x$.
$\begin{align}
& f\left( x \right)=\frac{x-2}{{{x}^{2}}-4} \\
& f\left( -x \right)=\frac{\left( -x \right)-2}{{{\left( -x \right)}^{2}}-4} \\
& =\frac{-\left( x+2 \right)}{{{x}^{2}}-4} \\
& \ne f\left( x \right)
\end{align}$
And,
$\begin{align}
& -f\left( x \right)=-\left( \frac{x-2}{{{x}^{2}}-4} \right) \\
& \ne f\left( -x \right)
\end{align}$
Hence, the graph of the function is not symmetric about the $y$-axis and origin.
Step 2: To calculate the x intercepts equate $f\left( x \right)=0$. Which never happens for any value of x.
Step 3: To calculate the y intercepts evaluate $f\left( 0 \right)$.
$f\left( 0 \right)=\frac{1}{2}$
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.
$x=-2$
