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Grade 11 · unit review

Review: Algebra2: Rational Functions and Domain

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.

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Unit review questions
1. What inputs are excluded from (x+1)/(x²−49)?
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. Simplify (x²−49)/(x−7) and keep its original domain.
Need a hint?
  • A shared polynomial factor can be canceled only where it is nonzero.
  • Cancellation applies to factors, not terms of a sum.
3. Solve 7/x=1/2, retaining the restriction x≠0.
Need a hint?
  • Clearing a variable denominator requires excluding its zero before multiplying.
  • A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it.
4. A positive inverse-variation rule y=k/x passes through (7,8). Find y at x=14.
Need a hint?
  • Inverse variation has the form y=k/x.
  • Inverse variation maintains a constant product rather than a constant difference or ratio y/x.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. 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. 2. Simplify (x²−81)/(x−9) and keep its original domain. Candidate answers: (A) x+81, with x≠9 (B) x+9, for every real x (C) x+9, with x≠9 (D) x−9, with x≠9. 3. Solve 9/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. 4. A positive inverse-variation rule y=k/x passes through (9,10). Find y at x=18. Candidate answers: (A) 10 (B) 4 (C) 5 (D) 6.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 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. 2. x+9, with x≠9. Factor the numerator and cancel the common nonzero factor x−9. The new formula is x+9, but the original exclusion x=9 remains; cancellation cannot fill that missing input. 3. 18. Multiply by 2x, permitted only for x≠0: 2(9)=x. The solution 18 is nonzero and substitutes to 9/18=1/2. 4. 5. First k=xy=9×10=90. Then y=90/18=5. Doubling x halves y; inverse variation is not a constant-difference rule.

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