Unit 3 Review and Quiz
A free Precalculus lesson from the “Inverse Functions” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint verifies that inverse relationships are solid before polynomial and rational work. 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 inverse meaning, one-to-one behavior, inverse rules, and inverse graphs
- Choose the correct function, graph, or modeling tool from mixed prompts
- Explain why the selected method fits the problem
Worked example
Problem. If f(4) = 9, what must f inverse(9) equal?
- Worked Example: First identify exactly what the question is asking: If f(4) = 9, what must f inverse(9) equal?
- For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
- An inverse reverses the input-output pair.
- f sends 4 to 9.
- The inverse sends 9 back to 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4
Practice problems
1. Unit review 1 (What an Inverse Function Means): If f(4) = 9, what must f inverse(9) equal?
Choices: 4 · 9 · 13 · -4
Show solution
- Unit Review: First identify exactly what the question is asking: If f(4) = 9, what must f inverse(9) equal?
- For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
- An inverse reverses the input-output pair.
- f sends 4 to 9.
- The inverse sends 9 back to 4.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4
2. Unit review 2 (One-to-One Functions): Which table could be one-to-one?
Choices: x: 1, 2, 3; y: 4, 5, 6 · x: 1, 2, 3; y: 4, 4, 6 · x: 1, 2, 3; y: 7, 7, 7 · x: 1, 1, 2; y: 3, 4, 5
Show solution
- One-to-one means different inputs have different outputs.
- Only the first table has no repeated output for different inputs.
- A repeated output would make the inverse fail the function rule.
Answer: x: 1, 2, 3; y: 4, 5, 6
3. Unit review 3 (Horizontal Line Test): A graph passes the horizontal line test when:
Choices: every horizontal line hits it at most once · every vertical line hits it twice · it crosses the y-axis · it has no x-intercepts
Show solution
- The horizontal line test checks one-to-one behavior.
- At most one hit means each output came from at most one input.
- Then the inverse can be a function.
Answer: every horizontal line hits it at most once
4. Unit review 4 (Finding Inverses Algebraically): Find the inverse of f(x) = (x - 5)/2. Enter in terms of x.
Show solution
- Unit Review: First identify exactly what the question is asking: Find the inverse of f(x) = (x - 5)/2. Enter in terms of x.
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Write y = (x - 5)/2.
- Multiply by 2: 2y = x - 5.
- Swap and solve to get f inverse(x) = 2x + 5.
- Check the result by substituting or estimating: the response should match 2x + 5 and make sense in the original problem.
Answer: 2x + 5
5. Unit review 5 (Graphs of Inverse Functions): A point that lies on both a function and its inverse often lies on:
Choices: y = x · x = 0 · y = 0 · a vertical asymptote
Show solution
- Points on y = x do not change when coordinates are swapped.
- That makes them natural intersection candidates.
- Other intersections can happen, but y = x is the reflection line.
Answer: y = x
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