Study guide: 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.
Ideas to explain in your own words
- The domain of a rational expression excludes inputs making its denominator zero. Factoring that denominator converts the exclusion search into solving simpler factor equations. A numerator zero is normally a valid zero output when the denominator stays nonzero.
- A shared polynomial factor can be canceled only where it is nonzero. The simpler formula agrees with the original expression on its original domain, which may retain a missing point. Cancellation applies to factors, not terms of a sum.
- Clearing a variable denominator requires excluding its zero before multiplying. Solving the transformed equation gives a candidate, and substitution verifies that it remains allowed and satisfies the original fraction. A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it.
- Inverse variation has the form y=k/x. A supplied nonzero pair determines the constant product xy=k, and a new input recovers its output by division. Inverse variation maintains a constant product rather than a constant difference or ratio y/x.
Useful vocabulary
- rational expression
- A quotient of polynomials with denominator nonzero.
- domain exclusion
- An input not allowed by the original formula.
- inverse variation
- A relationship y=k/x with constant k and x≠0.
Watch your reasoning
- 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.
- Cancellation applies to factors, not terms of a sum. Retain every original denominator exclusion in the simplified answer; a new formula without that restriction would represent a different domain.
- A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it. Record restrictions first so simplification does not accidentally hide them.
A worked review example
Complete supplied guide examples: Exclude Every Denominator Zero — 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. 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. Cancel Factors while Keeping the Hole — Simplify (x²−4)/(x−2) and keep its original domain. Candidate answers: (A) x+2, for every real x (B) x+2, with x≠2 (C) x+4, with x≠2 (D) x−2, with x≠2. Factor the numerator and cancel the common nonzero factor x−2. The new formula is x+2, but the original exclusion x=2 remains; cancellation cannot fill that missing input. Solve after Recording a Nonzero Variable Restriction — Solve 2/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Multiply by 2x, permitted only for x≠0: 2(2)=x. The solution 4 is nonzero and substitutes to 2/4=1/2. Use a Constant Product for Inverse Variation — A positive inverse-variation rule y=k/x passes through (2,3). Find y at x=4. Candidate answers: (A) 0.5 (B) 2.5 (C) 3 (D) 3/2. First k=xy=2×3=6. Then y=6/4=3/2. Doubling x halves y; inverse variation is not a constant-difference rule.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.