Precalculus Final Exam
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 exam 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 mastery 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 pairs each input with exactly one output.
- In {(2, 3), (3, 4), (4, 5)} no input repeats, so every input has a single output.
- Repeated outputs are fine; repeated inputs with different outputs are not.
Answer: {(2, 3), (3, 4), (4, 5)}
Practice problems
1. Final exam 1 (What a Function Is): Which relation is a function?
Choices: {(2, 3), (3, 4), (4, 5)} · {(2, 3), (2, 4), (4, 5)} · {(3, 3), (3, 5), (4, 4)} · {(4, 3), (4, 4), (2, 5)}
Show solution
- A function pairs each input with exactly one output.
- In {(2, 3), (3, 4), (4, 5)} no input repeats, so every input has a single output.
- Repeated outputs are fine; repeated inputs with different outputs are not.
Answer: {(2, 3), (3, 4), (4, 5)}
2. Final exam 2 (Function Notation): If f(x) = 4x - 3, find f(-3).
Show solution
- Full-Course Review: First identify exactly what the question is asking: If f(x) = 4x - 3, find f(-3).
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Substitute x = -3.
- 4(-3) - 3 = -12 - 3.
- So f(-3) = -15.
- Check the result by substituting or estimating: the response should match -15 and make sense in the original problem.
Answer: -15
3. Final exam 3 (Domain and Range from Graphs): A hollow (open) endpoint on a graph means that x-value is:
Choices: not included in the domain · included in the domain · the maximum output · the y-intercept
Show solution
- An open circle marks a value the graph approaches but does not reach.
- It is excluded from the domain.
- A filled circle would include it.
Answer: not included in the domain
4. Final exam 4 (Domain and Range from Equations): What is the domain of 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 of 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 a nonnegative radicand.
- Require 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 exam 5 (Intercepts and Key Features): The graph of y = (x - 2)^2 has its vertex at what x-value?
Show solution
- Full-Course Review: First identify exactly what the question is asking: The graph of y = (x - 2)^2 has its vertex at what x-value?
- For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
- A squared term is smallest when its inside is 0.
- x - 2 = 0 gives x = 2.
- So the vertex is at x = 2.
- Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.
Answer: 2
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