The Life Starter
Sign up
Grade 11 · lesson 1 of 4

Exclude Every Denominator Zero

The domain of a rational expression excludes inputs making its denominator zero.

About 25 minutes, plus any extended practice

Download the grade pack

Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: What inputs are excluded from (x+1)/(x²−4)? Candidate answers: (A) No inputs are excluded (B) x=2 only (C) x=−1 only (D) x=−2 and x=2.
  • Independent check inputs: What inputs are excluded from (x+1)/(x²−9)? Candidate answers: (A) No inputs are excluded (B) x=3 only (C) x=−1 only (D) x=−3 and x=3.

Learn the idea

Rational expressions divide polynomials, so denominator zeros are excluded from their domains. Factoring can reveal cancellation or help solve equations, but it does not erase the original restrictions or turn inverse variation into linear change. The domain of a rational expression excludes inputs making its denominator zero. Factoring that denominator converts the exclusion search into solving simpler factor equations.

Here is the method on the supplied teaching case. The denominator factors as (x−2)(x+2). Either zero factor makes division undefined, so exclude both 2 and −2. Numerator zeros do not cause a domain exclusion.

A numerator zero is normally a valid zero output when the denominator stays nonzero. Check every denominator factor; excluding only the positive solution of a squared denominator misses another forbidden input.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: What inputs are excluded from (x+1)/(x²−4)? Candidate answers: (A) No inputs are excluded (B) x=2 only (C) x=−1 only (D) x=−2 and x=2. Solution: The denominator factors as (x−2)(x+2). Either zero factor makes division undefined, so exclude both 2 and −2. Numerator zeros do not cause a domain exclusion. The final answer is x=−2 and x=2. A numerator zero is normally a valid zero output when the denominator stays nonzero. Check every denominator factor; excluding only the positive solution of a squared denominator misses another forbidden input.

Your turn

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Helpful tips
  • The domain of a rational expression excludes inputs making its denominator zero.
  • A numerator zero is normally a valid zero output when the denominator stays nonzero.
What inputs are excluded from (x+1)/(x²−9)?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

x=−3 and x=3

The denominator factors as (x−3)(x+3). Either zero factor makes division undefined, so exclude both 3 and −3. Numerator zeros do not cause a domain exclusion.

Take it further

Changed-input task: What inputs are excluded from (x+1)/(x²−81)? Candidate answers: (A) No inputs are excluded (B) x=9 only (C) x=−1 only (D) x=−9 and x=9. Show the mathematical reason, not only the selected answer. Expected result: x=−9 and x=9. The denominator factors as (x−9)(x+9). Either zero factor makes division undefined, so exclude both 9 and −9. Numerator zeros do not cause a domain exclusion.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. What inputs are excluded from (x+1)/(x²−16)?
Need a hint?
  • The domain of a rational expression excludes inputs making its denominator zero.
  • A numerator zero is normally a valid zero output when the denominator stays nonzero.
2. What inputs are excluded from (x+1)/(x²−25)?
Need a hint?
  • The domain of a rational expression excludes inputs making its denominator zero.
  • A numerator zero is normally a valid zero output when the denominator stays nonzero.
3. What inputs are excluded from (x+1)/(x²−36)?
Need a hint?
  • The domain of a rational expression excludes inputs making its denominator zero.
  • A numerator zero is normally a valid zero output when the denominator stays nonzero.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

Check the subject sources

These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.