Find all vertical asymptotes and holes of each function.
$f\left(x\right)=\frac{x^2-2x}{2x^3-x^2-6x}$

• A. Hole(s): None, Vertical Asymptote(s): $x=0,x=2,x=-\frac32$
• B. Hole(s): $x=0$, Vertical Asymptote(s): $x=-\frac32$
• C. Hole(s): $x=0$, $x=2$, Vertical Asymptote(s): $x=-\frac32$
• D. Hole(s): $x=0$, $x=2$, Vertical Asymptote(s): $x=\frac32$

Answer: (C)

Verified step by step guidance

1. Factor the numerator and the denominator of the function f(x) = \( \frac{x^2 - 2x}{2x^3 - x^2 - 6x} \).
2. Identify common factors in the numerator and the denominator. These common factors will indicate the location of holes in the graph.
3. Cancel out the common factors from the numerator and the denominator to simplify the function.
4. Set the remaining factors in the denominator equal to zero to find the vertical asymptotes. These are the x-values where the function is undefined and not canceled out.
5. Verify the x-values found for holes and vertical asymptotes by substituting them back into the simplified function to ensure they correspond to the correct behavior of the graph.