Solved Examples: Finding Vertical Asymptotes and Holes in Rational Functions

📚 Quick Study Guide

• 🔍 Rational Function: A function in the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
• ➗ Vertical Asymptotes: Occur where the denominator $Q(x) = 0$, but the factor does not cancel with a factor in the numerator $P(x)$. Solve for $x$ to find the vertical asymptote(s).
• 🕳️ Holes (Removable Discontinuities): Occur where a factor in the denominator $Q(x)$ also appears in the numerator $P(x)$ and can be cancelled out. To find the $x$-coordinate of the hole, set the cancelled factor equal to zero and solve for $x$. To find the $y$-coordinate, plug the $x$-value into the simplified function.
• 📝 Simplifying: Always simplify the rational function first by factoring both the numerator and the denominator.
• 📈 Graphing: Vertical asymptotes are vertical lines that the graph approaches but never crosses. Holes are points where the graph is undefined.

Practice Quiz

1. Question 1: What are the vertical asymptotes of the function $f(x) = \frac{x+2}{x^2 - 4}$?
   1. x = 2
   2. x = -2
   3. x = 2, x = -2
   4. x = 0
2. Question 2: What is the hole in the function $f(x) = \frac{x^2 - 9}{x - 3}$?
   1. x = -3
   2. x = 3
   3. There is no hole.
   4. x = 9
3. Question 3: What are the vertical asymptotes of the function $f(x) = \frac{x}{x^2 + 1}$?
   1. x = 1
   2. x = -1
   3. x = 1, x = -1
   4. There are no vertical asymptotes.
4. Question 4: What is the hole in the function $f(x) = \frac{x^2 - 4x + 4}{x - 2}$?
   1. x = 2
   2. x = -2
   3. There is no hole.
   4. x = 4
5. Question 5: What are the vertical asymptotes of the function $f(x) = \frac{x-1}{x^2 - 1}$?
   1. x = 1
   2. x = -1
   3. x = 1, x = -1
   4. There are no vertical asymptotes.
6. Question 6: What is the hole in the function $f(x) = \frac{x^2 - 5x + 6}{x - 2}$?
   1. x = 2
   2. x = 3
   3. There is no hole.
   4. x = -3
7. Question 7: What are the vertical asymptotes of the function $f(x) = \frac{x+5}{x^2 + 4x - 5}$?
   1. x = -5
   2. x = 1
   3. x = -5, x = 1
   4. x = -1, x = 5