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
- Review restrictions, holes, asymptotes, graphing, and rational equations
- Choose the correct function, graph, or modeling tool from mixed prompts
- Explain why the selected method fits the problem
Worked example
Problem. What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- Worked Example: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 3 = 0.
- The excluded value is 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
Practice problems
1. Unit review 1 (Rational Function Basics): What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
Show solution
- Unit Review: First identify exactly what the question is asking: What value is excluded from the domain of r(x) = (x + 1)/(x - 3)?
- For domain questions, identify input values that are allowed and watch for denominators, radicals, and context restrictions.
- The denominator cannot be zero.
- Set x - 3 = 0.
- The excluded value is 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
2. Unit review 2 (Domain Restrictions and Holes): In r(x) = (x - 3)/((x - 3)(x + 5)), what occurs at x = 3?
Choices: a hole · a vertical asymptote · a horizontal asymptote · no restriction
Show solution
- The factor x minus the same value cancels.
- Canceled restrictions become holes.
- Uncanceled denominator factors become vertical asymptotes.
Answer: a hole
3. Unit review 3 (Vertical Asymptotes): Find the vertical asymptote x-value for r(x) = 3/(x - (4)).
Show solution
- Unit Review: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (4)).
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Set the denominator equal to 0.
- x - (4) = 0 gives x = 4.
- Because no factor cancels, this is a vertical asymptote.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
4. Unit review 4 (Horizontal and Slant Asymptotes): Find the slant asymptote of r(x) = (x^2 + 2)/(x - 1) by long division. Enter it as y = mx + b.
Show solution
- The numerator degree is one more than the denominator, so expect a slant asymptote.
- Long division of x^2 + 2 by x - 1 gives quotient x + 1 (remainder 3).
- The slant asymptote is the quotient line y = x + 1.
Answer: y = x + 1
5. Unit review 5 (Graphing Rational Functions): A sign chart for a rational graph helps decide:
Choices: whether each interval is above or below the x-axis · the exact font size of labels · the course order · the value of pi
Show solution
- Intervals are split by zeros and restrictions.
- Testing signs tells whether the graph is positive or negative.
- That helps place branches.
Answer: whether each interval is above or below the x-axis
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