Step-by-Step Examples: Locating Holes in Rational Function Expressions

📚 Quick Study Guide

🔍 Rational Function: A function that can be written as $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials. ✂️ Simplification: Factor both the numerator and denominator and cancel out any common factors. 🕳️ Hole (Removable Discontinuity): Occurs when a factor cancels out from both the numerator and the denominator. 📍 Location of the Hole: After simplification, set the canceled factor equal to zero and solve for $x$. This is the x-coordinate of the hole. 📈 Finding the y-coordinate: Plug the x-coordinate of the hole back into the simplified rational function to find the corresponding y-coordinate. The hole is at the point $(x, y)$. 🚫 Vertical Asymptotes: Occur where the denominator of the *simplified* rational function equals zero.

Practice Quiz

1. What indicates a hole in a rational function?
   1. A) A factor that remains only in the denominator.
   2. B) A factor that remains only in the numerator.
   3. C) A factor that cancels out from both the numerator and denominator.
   4. D) A constant term in the simplified function.


2. Given the function $f(x) = \frac{(x-2)(x+1)}{(x-2)}$, where is the hole located?
   1. A) x = -1
   2. B) x = 2
   3. C) x = -2
   4. D) There is no hole.


3. What is the first step in finding holes in a rational function?
   1. A) Find the vertical asymptotes.
   2. B) Factor the numerator and denominator.
   3. C) Set the denominator equal to zero.
   4. D) Graph the function.


4. If a hole exists at $x = 3$ in a rational function that simplifies to $g(x)$, how do you find the y-coordinate of the hole?
   1. A) Plug $x = 3$ into the original function.
   2. B) Plug $x = 3$ into $g(x)$.
   3. C) Find the limit as $x$ approaches 3 of $g(x)$.
   4. D) Set $g(x) = 0$.


5. Where is the hole located in the function $f(x) = \frac{(x+5)(x-3)}{(x-3)(x+2)}$?
   1. A) (-2, 0)
   2. B) (3, 8/5)
   3. C) (-5, 0)
   4. D) (3, 0)


6. Consider the function $f(x) = \frac{x^2 - 4}{x - 2}$. What is the y-coordinate of the hole?
   1. A) 0
   2. B) 2
   3. C) 4
   4. D) Undefined


7. Which of the following functions has a hole at x = 1?
   1. A) $f(x) = \frac{x+1}{x-1}$
   2. B) $f(x) = \frac{x-1}{x+1}$
   3. C) $f(x) = \frac{(x-1)^2}{x-1}$
   4. D) $f(x) = \frac{1}{x-1}$