Precalculus Final Readiness Check
A free Precalculus lesson from the “Parametric, Polar, Vectors, and Intro to Limits” unit, with a worked example and practice problems including step-by-step solutions.
The final readiness check samples the full course: function notation, domain and range, transformations, composition, inverses, polynomial and rational behavior, exponential and logarithmic models, sequences and series, parametric and polar coordinates, vectors, difference quotients, informal limits, and rate-of-change reasoning. 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
- Demonstrate readiness across the full Precalculus course before moving toward Calculus
- Choose the correct function, graph, or modeling tool from mixed prompts
- Explain why the selected method fits the problem
Worked example
Problem. Which relation is a function?
- A function assigns each input exactly one output.
- In {(-2, 7), (0, 7), (4, 1)}, no input is paired with two different outputs.
- Repeated outputs do not break the function rule.
Answer: {(-2, 7), (0, 7), (4, 1)}
Practice problems
1. Final readiness 1 (What a Function Is): Which relation is a function?
Choices: {(-2, 7), (0, 7), (4, 1)} · {(-2, 7), (0, 1), (-2, 3)} · {(2, 6), (2, 8), (9, 1)} · {(0, 0), (0, 2), (5, 5)}
Show solution
- A function assigns each input exactly one output.
- In {(-2, 7), (0, 7), (4, 1)}, no input is paired with two different outputs.
- Repeated outputs do not break the function rule.
Answer: {(-2, 7), (0, 7), (4, 1)}
2. Final readiness 2 (Function Notation): If f(x) = 4x + 0, find f(-1).
Show solution
- Full-Course Review: First identify exactly what the question is asking: If f(x) = 4x + 0, find f(-1).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Substitute -1 for x.
- Compute 4(-1) + 0.
- The output is -4.
- Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.
Answer: -4
3. Final readiness 3 (Domain and Range from Graphs): A graph has y-values from -1 to 6. What is the range?
Choices: [-1, 6] · [-5, 7] · [-5, 6] · [-1, 7]
Show solution
- Full-Course Review: First identify exactly what the question is asking: A graph has y-values from -1 to 6. What is the range?
- For range questions, identify the possible output values after the input restrictions and graph shape are considered.
- Range is the y-value interval.
- The lowest y is -1 and the highest y is 6.
- Read range bottom to top.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: [-1, 6]
4. Final readiness 4 (Domain and Range from Equations): What is the domain condition for f(x) = sqrt(x - 6)?
Choices: x >= 6 · x > 6 · x <= 6 · x != 6
Show solution
- Full-Course Review: First identify exactly what the question is asking: What is the domain condition for f(x) = sqrt(x - 6)?
- For radicals, separate perfect-square factors when simplifying and check whether the radicand has any restrictions.
- A square root needs the radicand to be nonnegative.
- x - 6 >= 0.
- So x >= 6.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: x >= 6
5. Final readiness 5 (Intercepts and Key Features): A local maximum is a point where the graph:
Choices: changes from increasing to decreasing nearby · crosses the y-axis · has no domain · must be below the x-axis
Show solution
- A local maximum is a nearby high point.
- The graph rises into it and falls after it.
- It does not have to be the highest point forever.
Answer: changes from increasing to decreasing nearby
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