Graphs of Inverse Functions
A free Precalculus lesson from the “Inverse Functions” unit, with a worked example and practice problems including step-by-step solutions.
A function and its inverse trade x- and y-coordinates, so their graphs reflect across the line y = x. 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
- Connect inverse graphs to reflection across y = x
- Use graphs of inverse functions in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. The graph of an inverse function is the reflection of the original graph across:
- Inverse points swap coordinates.
- Swapping x and y reflects across y = x.
- This visual check helps confirm inverse graphs.
Answer: y = x
Practice problems
1. The graph of an inverse function is the reflection of the original graph across:
Choices: y = x · the x-axis · the y-axis · x = 0
Show solution
- Inverse points swap coordinates.
- Swapping x and y reflects across y = x.
- This visual check helps confirm inverse graphs.
Answer: y = x
2. The graph of an inverse function is the reflection of the original graph across: (variation 2)
Choices: y = x · the x-axis · the y-axis · x = 0
Show solution
- Inverse points swap coordinates.
- Swapping x and y reflects across y = x.
- This visual check helps confirm inverse graphs.
Answer: y = x
3. If (3, 8) is on f, what point is on f inverse?
Show solution
- Core Practice: First identify exactly what the question is asking: If (3, 8) is on f, what point is on f inverse?
- For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
- Inverse points swap x- and y-coordinates.
- (3, 8) becomes (8, 3).
- That matches reflection across y = x.
- Check the result by substituting or estimating: the response should match (8, 3) and make sense in the original problem.
Answer: (8, 3)
4. If (3, 8) is on f, what point is on f inverse? (variation 2)
Show solution
- Core Practice: First identify exactly what the question is asking: If (3, 8) is on f, what point is on f inverse?
- For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
- Inverse points swap x- and y-coordinates.
- (3, 8) becomes (8, 3).
- That matches reflection across y = x.
- Check the result by substituting or estimating: the response should match (8, 3) and make sense in the original problem.
Answer: (8, 3)
5. 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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