Precalculus (6th Edition) Blitzer

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. $x=-2$