The problem you're working on asks to determine the vertical asymptotes and holes (removable discontinuities) of the given rational function:

$f(x) = \frac{x^2 - 12x + 20}{x^2 - 9x + 14}$

Step 1: Factor both the numerator and the denominator.

1. Numerator: $x^2 - 12x + 20$ Factors as: $(x - 10)(x - 2)$

2. Denominator: $x^2 - 9x + 14$ Factors as: $(x - 7)(x - 2)$

Step 2: Find the holes.

A hole occurs when a factor cancels out from both the numerator and denominator. In this case, the factor $(x - 2)$ appears in both, meaning there is a hole at $x = 2$.

Step 3: Find the vertical asymptotes.

Vertical asymptotes occur where the denominator is zero, but the factor doesn't cancel out. After canceling $(x - 2)$, we're left with: $\frac{x - 10}{x - 7}$ Thus, the vertical asymptote occurs at $x = 7$, because $x - 7 = 0$ when $x = 7$.

Final Answers:

• Hole: $x = 2$
• Vertical Asymptote: $x = 7$

Would you like more details, or do you have any further questions?

Related Questions:

1. What is the general process for identifying vertical asymptotes in rational functions?
2. How do you determine if a discontinuity is removable or non-removable?
3. What role does factoring play in finding holes and asymptotes of rational functions?
4. Can there be more than one hole or vertical asymptote in a rational function?
5. How does the degree of the numerator and denominator affect the graph of a rational function?

Tip:

Always check for common factors between the numerator and denominator first, as they may indicate a hole in the graph.