Precalculus Midterm Exam
A free Precalculus lesson from the “Polynomial Functions” unit, with a worked example and practice problems including step-by-step solutions.
The midterm exam mixes Units 1-4 without topic labels: function notation, domain and range, graph features, average rate of change, transformations, symmetry, operations, composition, inverse functions, and full polynomial behavior (degree, end behavior, zeros, multiplicity, division, and the factor theorem). 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 cumulative mastery across functions, transformations, inverses, and polynomial functions
- 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. Midterm 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. Midterm 2 (Function Notation): If f(x) = 4x - 3, find f(-3).
Show solution
- Units 1-2 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. Midterm 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. Midterm 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
- Units 1-2 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. Midterm 5 (Intercepts and Key Features): The graph of y = (x - 2)^2 has its vertex at what x-value?
Show solution
- Units 1-2 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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