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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

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

Worked example

Problem. The graph of f inverse is the reflection of the graph of f across which line?

  1. Inverse functions trade x- and y-coordinates.
  2. Swapping the coordinates of every point reflects the graph across y = x.
  3. So the axis of reflection is the line y = x.

Answer: y = x

Practice problems

1. The graph of f inverse is the reflection of the graph of f across which line?

Choices: y = x · the x-axis · the y-axis · x = 0

Show solution
  1. Inverse functions trade x- and y-coordinates.
  2. Swapping the coordinates of every point reflects the graph across y = x.
  3. So the axis of reflection is the line y = x.

Answer: y = x

2. The point (4, 11) lies on f. Give the matching point on f inverse as (x, y).

Show solution
  1. On f inverse the coordinates of each point are swapped.
  2. (4, 11) becomes (11, 4).
  3. Check: swapping (11, 4) back gives (4, 11), the original point on f.

Answer: (11, 4)

3. The point (5, 12) lies on f. On f inverse the matching point is (x, y). What is its x-coordinate?

Show solution
  1. Core Practice: First identify exactly what the question is asking: The point (5, 12) lies on f. On f inverse the matching point is (x, y). What is its x-coordinate?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Reflecting across y = x swaps the coordinates.
  4. (5, 12) becomes (12, 5).
  5. So the x-coordinate of the image is 12.
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

4. The point (6, 13) lies on f. On f inverse the matching point is (x, y). What is its y-coordinate?

Show solution
  1. Core Practice: First identify exactly what the question is asking: The point (6, 13) lies on f. On f inverse the matching point is (x, y). What is its y-coordinate?
  2. For inverse relationships, reverse the operations in the opposite order and check that the result undoes the original rule.
  3. Reflecting across y = x swaps the coordinates.
  4. (6, 13) becomes (13, 6).
  5. So the y-coordinate of the image is 6.
  6. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. 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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