Inverse functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Inverse functions

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. Which rule undoes f(t) = 4t - 8?

    • f⁻¹(x) = 4x + 8
    • f⁻¹(x) = (x + 8)/4
    • f⁻¹(x) = (x - 8)/4
    • f⁻¹(x) = x/4 - 8

  2. The point (1, 7) lies on an invertible function f. Which corresponding point lies on the graph of f⁻¹?

    • (1, 7)
    • (-1, -7)
    • (7, -1)
    • (7, 1)

  3. Which expression is the reciprocal of f(x) = 4x - 3?

    • (x + 3) / 4
    • 4x + 3
    • 1 / (4x - 3)
    • 1 / (4x) - 3

  4. An invertible function sends 6 to 15. Its inverse sends 15 to 6.

    Circle one:   True   False

  5. For f(t) = 7 - t on the real numbers, f⁻¹ has the same rule as f.

    Circle one:   True   False

  6. Find the inverse of f(t) = (t + 6)/3.

    • f⁻¹(x) = 3x + 6
    • f⁻¹(x) = (x - 6)/3
    • f⁻¹(x) = x/3 - 6
    • f⁻¹(x) = 3x - 6

  7. Leo finds the inverse of f(x) = x + 3 and writes f⁻¹(x) = 1/(x + 3), saying the -1 means reciprocal. Where is the slip?

    • Swap and rearrange to get x - 3
    • He should have squared the rule
    • The reciprocal is actually correct
    • He should have divided by 3

  8. The graph of the inverse of f is the reflection of the graph of f in the line y = x.

    Circle one:   True   False

  9. For f(x) = 3x + 1, what is f⁻¹(f⁻¹(22)), the inverse applied twice?

    • 22 / 9
    • 7
    • 8
    • 2

  10. The inverse of f(x) = 4x - 7 is (x + 7) / 4. Feed 5 through the inverse rule, then feed the result through it again. What do you get? Give your answer as a decimal.

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Inverse functions W1-mt_7k7q5ZjMb4-s1

  1. f⁻¹(x) = (x + 8)/4 · The inverse adds 8 and then divides by 4.
  2. (7, 1) · The inverse graph is the reflection in y = x, which trades each point's coordinates, so (1, 7) becomes (7, 1).
  3. 1 / (4x - 3) · The reciprocal is 1 divided by the whole rule: 1 / (4x - 3). The inverse would be (x + 3) / 4, which is different.
  4. True · The inverse reverses this input-output assignment.
  5. True · Swapping x = 7 - y and rearranging gives y = 7 - x, the original rule.
  6. f⁻¹(x) = 3x - 6 · Undo division by multiplying by 3, then undo addition by subtracting 6.
  7. Swap and rearrange to get x - 3 · The minus one is a label, not a power. Swapping gives x = y + 3, so y = x - 3, and that is the inverse, not 1 / (x + 3).
  8. True · Swapping x and y trades each point's coordinates, which is exactly the reflection in the line y = x.
  9. 2 · The inverse of 3x + 1 is (x - 1) / 3. Apply it: (22 - 1) / 3 = 7. Apply it again: (7 - 1) / 3 = 2.
  10. 2.5 · Apply the inverse: (5 + 7) / 4 = 3. Apply it again: (3 + 7) / 4 = 2.5.
Worksheet · LightMySky