CMClearMathAcademy

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

Why it matters: Rational functions model rates, constraints, efficiency, and quantities that change sharply near restricted inputs.

Worked example

Problem. What value of x is excluded from the domain of r(x) = (x + 1)/(x - 3)?

  1. A rational function is undefined where its denominator is 0.
  2. Set the denominator equal to 0: x - 3 = 0.
  3. Solving gives x = 3, so 3 is excluded from the domain.

Answer: 3

Practice problems

1. What value of x is excluded from the domain of r(x) = (x + 1)/(x - 3)?

Show solution
  1. A rational function is undefined where its denominator is 0.
  2. Set the denominator equal to 0: x - 3 = 0.
  3. Solving gives x = 3, so 3 is excluded from the domain.

Answer: 3

2. What value of x is excluded from the domain of r(x) = (x - 2)/(x + 4)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: What value of x is excluded from the domain of r(x) = (x - 2)/(x + 4)?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. The denominator cannot equal 0.
  4. Set x + 4 = 0.
  5. Solving gives x = -4, the only excluded value.
  6. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

3. What value of x is excluded from the domain of r(x) = (x + 3)/(2x - 10)?

Show solution
  1. Core Practice: First identify exactly what the question is asking: What value of x is excluded from the domain of r(x) = (x + 3)/(2x - 10)?
  2. For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
  3. Set the denominator to 0: 2x - 10 = 0.
  4. Add 10 to both sides: 2x = 10.
  5. Divide both sides by 2: x = 10/2 = 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. For r(x) = (x + 2)/(x - 6), evaluate r(7).

Show solution
  1. Core Practice: First identify exactly what the question is asking: For r(x) = (x + 2)/(x - 6), evaluate r(7).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Substitute x = 7: r(7) = (7 + 2)/(7 - 6).
  4. Numerator = 9; denominator = 1.
  5. r(7) = 9/1 = 9.
  6. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. Find the x-intercept (the x-value) of r(x) = (x - 5)/(x + 3).

Show solution
  1. An x-intercept occurs where r(x) = 0, i.e. where the numerator is 0 (and the denominator is not).
  2. Set the numerator to 0: x - 5 = 0, so x = 5.
  3. Check the denominator at x = 5: 5 + 3 = 8 (not 0), so the x-intercept is at x = 5.

Answer: 5

Practice this interactively with instant feedback and an AI tutor.

Practice Rational Function Basics Take the free placement check

More Precalculus lessons