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Holes, Intercepts, and Discontinuities
AP Precalculus • Unit 1
Holes, Intercepts & Discontinuities
Learn how to identify holes, x- and y-intercepts, and different types of discontinuities in rational functions using algebraic reasoning.
Lesson Objectives
- Identify holes in rational functions
- Find x- and y-intercepts
- Understand removable and non-removable discontinuities
- Connect algebraic form to graph behavior
1. Holes in Rational Functions
A hole occurs when a factor in the denominator cancels with the numerator.
Example (OpenStax style):
f(x) = (x − 2)(x + 3)/(x − 2)
f(x) = (x − 2)(x + 3)/(x − 2)
- Factor (x − 2) cancels
- Hole at x = 2
After simplification: f(x) = x + 3, but x ≠ 2
📌 Diagram placeholder: hole on rational function graph
2. Intercepts of Rational Functions
Intercepts show where a graph crosses the coordinate axes.
x-intercepts:
Set the numerator = 0 (after simplification).
Set the numerator = 0 (after simplification).
y-intercept:
Evaluate the function at x = 0, if defined.
Evaluate the function at x = 0, if defined.
Example:
f(x) = (x + 1)/(x − 2)
f(x) = (x + 1)/(x − 2)
- x-intercept: x + 1 = 0 → x = −1
- y-intercept: f(0) = −1/−2 = 1/2
📊 Diagram placeholder: intercepts on rational graph
3. Discontinuities
A function is discontinuous at values where it is undefined or has a break in its graph.
Removable Discontinuity
- Occurs at a hole
- Factor cancels
Non-Removable Discontinuity
- Vertical asymptote
- Denominator = 0 without cancellation
📈 Diagram placeholder: hole vs vertical asymptote
4. Putting It All Together
Example (AP-style):
f(x) = (x − 1)(x + 2)/(x − 1)(x − 3)
f(x) = (x − 1)(x + 2)/(x − 1)(x − 3)
- Hole at x = 1
- Vertical asymptote at x = 3
- x-intercept at x = −2
5. Summary Table
| Feature | How to Identify |
|---|---|
| Hole | Factor cancels |
| x-Intercept | Numerator = 0 |
| y-Intercept | x = 0 |
| Vertical Asymptote | Denominator = 0 (no cancel) |
Practice Questions
- Find any holes of f(x) = (x − 4)/(x − 4)(x + 1)
- Find the x-intercept of f(x) = (x − 3)/(x + 2)
- Is the discontinuity at x = −1 removable or non-removable?
✅ Show Answer Key
- Hole at x = 4
- x = 3
- Non-removable
© Aviate Learning – AP Precalculus