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What an Inverse Function Means

A free Precalculus lesson from the “Inverse Functions” unit, with a worked example and practice problems including step-by-step solutions.

An inverse reverses the input-output pairing of a function. 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. 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. f inverse(14) = 4. What is f(4)?

Show solution
  1. Warm-up: First identify exactly what the question is asking: f inverse(14) = 4. What is f(4)?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. If f inverse(14) = 4, then f reverses that pairing.
  4. f must send 4 back to 14.
  5. So f(4) = 14.
  6. Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.

Answer: 14

3. If f(5) = 16, what must f inverse(16) equal?

Choices: 5 · 16 · 17 · -5

Show solution
  1. Core Practice: First identify exactly what the question is asking: If f(5) = 16, what must f inverse(16) equal?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. An inverse reverses the input-output pair.
  4. f sends 5 to 16.
  5. So f inverse sends 16 back to 5.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 5

4. g(x) = 3x + 6. Solving y = 3x + 6 for x gives the inverse rule g inverse(y) = (y - 6) / 3. Use it to find g inverse(15).

Show solution
  1. Start from the inverse rule g inverse(y) = (y - 6) / 3.
  2. Substitute y = 15: g inverse(15) = (15 - 6) / 3 = 9 / 3.
  3. 9 / 3 = 3, so g inverse(15) = 3.

Answer: 3

5. Because f inverse undoes f, what is f(f inverse(3))?

Show solution
  1. Applying f inverse and then f returns the starting value.
  2. f(f inverse(x)) = x for every x in the range of f.
  3. So f(f inverse(3)) = 3.

Answer: 3

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