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. The point (1, 7) lies on the graph of f. Which point lies on the graph of f to the power of -1?

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

  2. Find the inverse of f(x) = 3x + 2.

    • (x - 3) / 2
    • (x - 2) / 3
    • (x + 2) / 3
    • 3x - 2

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

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

  4. For f(x) = 5x - 2, what is f to the power of -1, of 3?

    • 3
    • 1
    • 13
    • 25

  5. You are told f to the power of -1, of x, is 2x - 3. What is f(5)?

    Answer: ______________

  6. Dev finds the inverse of f(x) = 3x - 2. He swaps to get x = 3y - 2, then writes f to the power of -1, of x, as (x - 2) / 3. Where is the slip?

    • He should have added 2, then divided by 3, to get (x + 2) / 3
    • He should have divided by 2 instead of 3
    • The swap was done wrongly
    • 3x - 2 is its own inverse

  7. The inverse of f(x) = (2x + 1) / 3 is (3x + 1) / 2.

    Circle one:   True   False

  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. The point (a, b) lies on the graph of f, and the point (9, 3) lies on the graph of f to the power of -1. What is a + b?

    Answer: ______________

  10. For f(x) = 3x + 1, what is f to the power of -1, of f to the power of -1, of 22, that is the inverse applied twice?

    • 22 / 9
    • 7
    • 8
    • 2
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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. (7, 1) · The inverse graph is the reflection in y = x, which trades each point's coordinates, so (1, 7) becomes (7, 1).
  2. (x - 2) / 3 · Swap to get x = 3y + 2, then 3y = x - 2, so y = (x - 2) / 3.
  3. 1 / (4x - 3) · The reciprocal is 1 divided by the whole rule, which is 1 / (4x - 3). The inverse would be (x + 3) / 4, a different animal.
  4. 1 · The inverse of 5x - 2 is (x + 2) / 5. Feed in 3: (3 + 2) / 5 = 1.
  5. 4 · The inverse is 2x - 3, so f is its own inverse found by swapping: x = 2y - 3 gives y = (x + 3) / 2. Then f(5) = (5 + 3) / 2 = 4.
  6. He should have added 2, then divided by 3, to get (x + 2) / 3 · From x = 3y - 2, add 2 to get x + 2 = 3y, then divide by 3 to get y = (x + 2) / 3. Dev subtracted 2 instead of adding it.
  7. False · Swapping gives x = (2y + 1) / 3, and rearranging gives y = (3x - 1) / 2. The claimed rule has the wrong sign: it should be 3x - 1, not 3x + 1.
  8. True · Swapping x and y trades each point's coordinates, which is exactly the reflection in the line y = x.
  9. 12 · Reflection in y = x trades coordinates, so (a, b) on f means (b, a) on the inverse. Since (9, 3) is on the inverse, b = 9 and a = 3, so a + b = 12.
  10. 2 · The inverse of 3x + 1 is (x - 1) / 3. Apply it: (22 - 1) / 3 = 7. Apply it again: (7 - 1) / 3 = 2.
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