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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

Why it matters: Inverse relationships show up in logarithms, unit conversions, solving formulas, and undoing a process.

Worked example

Problem. f(3) = 12. What is f inverse(12)?

  1. f inverse undoes f, so it sends an output back to its input.
  2. f sends 3 to 12, so f inverse sends 12 back to 3.
  3. Check: f(3) = 12, so f inverse(12) = 3.

Answer: 3

Practice problems

1. Unit review 1 (What an Inverse Function Means): f(3) = 12. What is f inverse(12)?

Show solution
  1. f inverse undoes f, so it sends an output back to its input.
  2. f sends 3 to 12, so f inverse sends 12 back to 3.
  3. Check: f(3) = 12, so f inverse(12) = 3.

Answer: 3

2. Unit review 2 (One-to-One Functions): Which table could come from a one-to-one function?

Choices: x: 3, 4, 5; y: 5, 7, 9 · x: 3, 4, 5; y: 5, 5, 9 · x: 3, 4, 5; y: 7, 7, 7 · x: 3, 4, 5; y: 9, 5, 9

Show solution
  1. One-to-one means no output repeats for different inputs.
  2. Only x: 3, 4, 5; y: 5, 7, 9 lists three different outputs.
  3. Any repeated output would break one-to-one.

Answer: x: 3, 4, 5; y: 5, 7, 9

3. Unit review 3 (Horizontal Line Test): A horizontal line drawn at height y = 5 crosses the parabola y = x^2 in how many points?

Show solution
  1. Solve x^2 = 5 for the crossings.
  2. Since 5 > 0, there are two solutions: x = sqrt(5) and x = -sqrt(5).
  3. So the horizontal line meets the parabola in 2 points.

Answer: 2

4. Unit review 4 (Finding Inverses Algebraically): Find the inverse of f(x) = x - 5. Enter f inverse(x) in terms of x.

Show solution
  1. Unit Review: First identify exactly what the question is asking: Find the inverse of f(x) = x - 5. Enter f inverse(x) in terms of x.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Write y = x - 5.
  4. Swap x and y: x = y - 5.
  5. Solve for y: y = x + 5.
  6. Check the result by substituting or estimating: the response should match x + 5 and make sense in the original problem.

Answer: x + 5

5. Unit review 5 (Graphs of Inverse Functions): The point (7, 9) lies on the graph of f. Which point must lie on the graph of f inverse?

Choices: (9, 7) · (7, 9) · (-9, -7) · (9, -7)

Show solution
  1. f inverse reverses each input-output pair.
  2. So (7, 9) on f becomes (9, 7) on f inverse.
  3. The forward function sends 7 to 9, and the inverse sends 9 back to 7.

Answer: (9, 7)

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