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
- Explain inverse functions as relationships that undo each other
- Use what an inverse function means in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
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. If f(4) = 9, what must f inverse(9) equal?
Choices: 4 · 9 · 13 · -4
Show solution
- Warm-up: 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. If f(4) = 9, what must f inverse(9) equal? (variation 2)
Choices: 4 · 9 · 13 · -4
Show solution
- Warm-up: 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
3. Two functions are inverses when their composition gives:
Choices: the original input · the original output plus 1 · a vertical shift · zero for every input
Show solution
- One function undoes the other.
- After both are applied, the input should return.
- That is why f(g(x)) = x and g(f(x)) = x.
Answer: the original input
4. Two functions are inverses when their composition gives: (variation 2)
Choices: the original input · the original output plus 1 · a vertical shift · zero for every input
Show solution
- One function undoes the other.
- After both are applied, the input should return.
- That is why f(g(x)) = x and g(f(x)) = x.
Answer: the original input
5. The inverse of a function swaps:
Choices: inputs and outputs · slopes and y-intercepts only · positive and negative signs only · domain and coefficients only
Show solution
- Core Practice: First identify exactly what the question is asking: The inverse of a function swaps:
- For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
- Inputs become outputs.
- Outputs become inputs.
- This is why inverse graphs reflect across y = x.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: inputs and outputs
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