A reliable method
Factor numerator and denominator, record exclusions, then simplify. Find intercepts only at allowed inputs. Compare numerator and denominator degrees before discussing horizontal behavior.
The common trap
Canceling a common factor creates a simpler formula on the remaining domain; it does not fill the original hole.
Make the connection
A vertical asymptote and a removable hole are different features. Use the unfactored domain and the simplified expression together.
A hole survives cancellation
R(x)=(x²−9)/(x−3) agrees with x+3 only for x≠3. At x=5 its value is 8, but at x=3 the original quotient is undefined. Compare S(x)=1/(x−3), whose output grows without bound near the excluded input: this is a vertical asymptote rather than a removable hole. Cancellation indicates which behavior to investigate; recording the original restriction prevents a simplified formula from changing the function. For a horizontal asymptote, compare leading terms instead of confusing an excluded input with a limiting output. Domain, zero and asymptote height are different quantities and should be labeled.
Transfer, explain and check
For T(x)=(x²−4)/(x−2), find T(0), the excluded input and the hole’s height. Check: 2, x=2 excluded, hole at (2,4). The hole height is not an actual function value.
For R(x)=(x²−9)/(x−3), which input is excluded from its original domain?
- A denominator cannot be zero.
- Solve x−3=0.
- The input 3 remains excluded even though the expression simplifies.
Compare the other answer choices
Use the cancelled expression’s value rather than the denominator zero.
Read the squared constant as the forbidden input.
Use the other numerator zero as the exclusion.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus course and syllabus