Rational Function Basics
A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.
A rational function is a quotient of polynomials, so denominator zeros control restrictions. 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
- Identify numerator, denominator, domain restrictions, and basic rational behavior
- Use rational function basics in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- Worked Example: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 3 = 0.
- The excluded value is 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
Practice problems
1. What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
Show solution
- Warm-up: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 3 = 0.
- The excluded value is 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
2. What value is excluded from the domain of r(x) = (x + 1)/(x - 4)?
Show solution
- Warm-up: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 4)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 4 = 0.
- The excluded value is 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
3. What value is excluded from the domain of r(x) = (x + 1)/(x - 5)?
Show solution
- Core Practice: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 5)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 5 = 0.
- The excluded value is 5.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
4. A rational function is formed by:
Choices: a quotient of polynomials · a sum of two logarithms · only a square root · a sequence with no formula
Show solution
- Rational functions have polynomial numerator and denominator.
- The denominator cannot be the zero polynomial.
- Restrictions come from denominator zeros.
Answer: a quotient of polynomials
5. A rational function is formed by: (variation 2)
Choices: a quotient of polynomials · a sum of two logarithms · only a square root · a sequence with no formula
Show solution
- Rational functions have polynomial numerator and denominator.
- The denominator cannot be the zero polynomial.
- Restrictions come from denominator zeros.
Answer: a quotient of polynomials
Practice this interactively with instant feedback and an AI tutor.
Practice Rational Function Basics Take the free placement check