Detailed Examples of Finding Holes in Rational Function Equations

📚 Quick Study Guide

• 🔎 A rational function is a function that can be written as the ratio of two polynomials, i.e., $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials.
• ✂️ A hole (or removable discontinuity) occurs in a rational function when a factor cancels out from both the numerator and the denominator.
• ✍️ To find holes: Factor both the numerator and the denominator.
• 📉 Identify any common factors.
• ➗ Cancel the common factors.
• 📍 Set the canceled factor equal to zero and solve for $x$. This $x$-value is the $x$-coordinate of the hole.
• 📈 Substitute this $x$-value into the simplified function to find the $y$-coordinate of the hole.
• 💡 The hole is then located at the point $(x, y)$.

🧪 Practice Quiz

1. What indicates a hole in a rational function?
   1. A) A factor that appears only in the numerator.
   2. B) A factor that appears only in the denominator.
   3. C) A factor that cancels out from both the numerator and denominator.
   4. D) A factor that does not appear at all.


2. Given the function $f(x) = \frac{(x-2)(x+3)}{(x-2)(x-1)}$, where is the hole located?
   1. A) $(2, 5)$
   2. B) $(2, -5)$
   3. C) $(2, 0)$
   4. D) There is no hole.


3. What is the first step in finding holes in a rational function?
   1. A) Simplify the function.
   2. B) Factor the numerator and the denominator.
   3. C) Solve for $x$.
   4. D) Find the y-intercept.


4. If a factor of $(x+5)$ cancels out in a rational function, what is the x-coordinate of the hole?
   1. A) $5$
   2. B) $-5$
   3. C) $0$
   4. D) $1$


5. What do you do after finding the x-coordinate of the hole?
   1. A) Nothing, you're done.
   2. B) Substitute it into the original function.
   3. C) Substitute it into the simplified function.
   4. D) Set the denominator equal to zero.


6. In the function $f(x) = \frac{(x-4)}{(x-4)(x+2)}$, what is the y-coordinate of the hole?
   1. A) $1/6$
   2. B) $6$
   3. C) $-2$
   4. D) There is no hole.


7. What is the location of the hole in the rational function $f(x) = \frac{(x+1)(x-2)}{(x-2)}$?
   1. A) $(2, 3)$
   2. B) $(-1, 0)$
   3. C) $(2, 0)$
   4. D) There is no hole.