Domain Restrictions and Holes
A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.
A canceled factor creates a hole; an uncanceled denominator factor creates a vertical asymptote. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.
What you'll learn
- Distinguish excluded values that cancel from vertical asymptotes that remain
- Use domain restrictions and holes in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. In r(x) = (x - 2)/((x - 2)(x + 4)), what occurs at x = 2?
- The factor x minus the same value cancels.
- Canceled restrictions become holes.
- Uncanceled denominator factors become vertical asymptotes.
Answer: a hole
Practice problems
1. In r(x) = (x - 2)/((x - 2)(x + 4)), what occurs at x = 2?
Choices: a hole · a vertical asymptote · a horizontal asymptote · no restriction
Show solution
- The factor x minus the same value cancels.
- Canceled restrictions become holes.
- Uncanceled denominator factors become vertical asymptotes.
Answer: a hole
2. In r(x) = (x - 3)/((x - 3)(x + 5)), what occurs at x = 3?
Choices: a hole · a vertical asymptote · a horizontal asymptote · no restriction
Show solution
- The factor x minus the same value cancels.
- Canceled restrictions become holes.
- Uncanceled denominator factors become vertical asymptotes.
Answer: a hole
3. In r(x) = (x - 4)/((x - 4)(x + 6)), what occurs at x = 4?
Choices: a hole · a vertical asymptote · a horizontal asymptote · no restriction
Show solution
- The factor x minus the same value cancels.
- Canceled restrictions become holes.
- Uncanceled denominator factors become vertical asymptotes.
Answer: a hole
4. After simplifying (x - 1)/((x - 1)(x + 3)), what x-value marks the hole?
Show solution
- Core Practice: First identify exactly what the question is asking: After simplifying (x - 1)/((x - 1)(x + 3)), what x-value marks the hole?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- The factor x - 1 cancels.
- The original denominator was zero at x = 1.
- That canceled restriction is a hole at x = 1.
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
Answer: 1
5. After simplifying (x - 2)/((x - 2)(x + 4)), what x-value marks the hole?
Show solution
- Core Practice: First identify exactly what the question is asking: After simplifying (x - 2)/((x - 2)(x + 4)), what x-value marks the hole?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- The factor x - 2 cancels.
- The original denominator was zero at x = 2.
- That canceled restriction is a hole at x = 2.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
Practice this interactively with instant feedback and an AI tutor.
Practice Domain Restrictions and Holes Take the free placement check