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Unit 5 Review and Quiz

A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.

This checkpoint verifies rational-function behavior before exponential and logarithmic models. 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. Unit review 1 (Rational Function Basics): 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. Unit review 2 (Domain Restrictions and Holes): For r(x) = (3(x - 4))/(x - 4), cancel the shared (x - 4) factor and find the y-value of the hole.

Show solution
  1. The factor (x - 4) appears in both the numerator and the denominator, so cancel it.
  2. What remains is the constant 3, defined for every x except 4.
  3. The hole sits on that constant value, so its y-value is 3.

Answer: 3

3. Unit review 3 (Vertical Asymptotes): r(x) = 5/((x - 5)(x - 12)). Enter the LARGER of the two vertical-asymptote x-values.

Show solution
  1. Each denominator factor set to 0 gives a vertical asymptote.
  2. x - 5 = 0 gives x = 5; x - 12 = 0 gives x = 12.
  3. The larger value is x = 12.

Answer: 12

4. Unit review 4 (Horizontal and Slant Asymptotes): Find the slant asymptote of r(x) = (x^2 + 3x + 9)/(x + 1) by long division. Enter it as y = mx + b.

Show solution
  1. Degrees differ by one (2 vs 1), so there is a slant asymptote equal to the quotient.
  2. Dividing: x^2 + 3x + 9 = (x + 1)(x + 2) + 7, so the quotient is x + 2 with a nonzero remainder.
  3. The slant asymptote is y = x + 2.

Answer: y = x + 2

5. Unit review 5 (Graphing Rational Functions): Find the vertical asymptote x-value for r(x) = 4/(x - 3).

Show solution
  1. A vertical asymptote occurs where the uncanceled denominator is 0.
  2. Set x - 3 = 0, so x = 3; the numerator 4 does not cancel it.
  3. So the vertical asymptote is x = 3.

Answer: 3

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