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

Composite functions

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

  1. The rules are f(x) = 2x and g(x) = x + 6. Here fg(x) means f(g(x)). What is fg(1)?

    • 8
    • 14
    • 4
    • 16

  2. In the composite f(g(x)), the rule f is applied first.

    Circle one:   True   False

  3. The rules are f(x) = x - 3 and g(x) = 2x. Here fg(x) means f(g(x)). What is fg(4)?

    • 8
    • 2
    • 5
    • 14

  4. With f(t) = 3t - 2 and g(t) = t + 5, let gf(t) mean g(f(t)). Calculate gf(4).

    Answer: ______________

  5. The rules are f(x) = x + 1 and g(x) = 2x + 1. Here fg(x) means f(g(x)) and gf(x) means g(f(x)). What are fg(2) and gf(2)?

    • 6 and 7
    • 6 and 6
    • 7 and 7
    • 7 and 6

  6. For f(x) = 2x and g(x) = x squared, let gf(x) mean g(f(x)). The value of gf(3) is 36.

    Circle one:   True   False

  7. For the rules f(x) = x + 2 and g(x) = 2x, the composites fg(x) and gf(x) are the same rule.

    Circle one:   True   False

  8. Here fg(x) means f(g(x)). Maya finds fg(2) for f(x) = 2x + 3 and g(x) = x + 1. She starts with f, gets f(2) = 7, then feeds 7 into g and writes fg(2) = 8. Where is the slip?

    • fg(2) should indeed be 8
    • She used f first; fg starts with g
    • She should have added the two outputs
    • She should have multiplied the two outputs

  9. The rules are f(x) = x + 2 and g(x) = 4x. Here gf(x) means g(f(x)) and fg(x) means f(g(x)). What is gf(2) - fg(2)?

    Answer: ______________

  10. The rules are f(x) = x squared and g(x) = x - 2. Here fg(x) means f(g(x)) and gf(x) means g(f(x)). What is fg(4) - gf(4)?

    Answer: ______________

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Answer key

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

Composite functions W1-mt_jqGJHXgYNw-s1

  1. 14 · g acts first: g(1) is 7. Feed that into f: f(7) is 2 times 7 = 14.
  2. False · In f(g(x)), g(x) is the inner input. Apply g first, then apply f to its output.
  3. 5 · g acts first: g(4) is 8. Feed that into f: f(8) is 8 - 3 = 5.
  4. 15 · First f(4) = 10. Then g(10) = 15.
  5. 6 and 7 · fg(2): g(2) is 5, f(5) is 6. gf(2): f(2) is 3, g(3) is 7. So fg(2) is 6 and gf(2) is 7.
  6. True · In gf, f acts first: f(3) is 6. Feed that into g: g(6) is 6 squared = 36.
  7. False · fg(x) = f(2x) = 2x + 2, and gf(x) = g(x + 2) = 2x + 4. They differ by 2, so they are not the same rule.
  8. She used f first; fg starts with g · In fg, g acts first: g(2) is 3, and f(3) is 2(3) + 3 = 9. Maya ran the pipeline in the wrong order, which gives gf(2) = 8.
  9. 6 · gf(2): f(2) is 4, g(4) is 16. fg(2): g(2) is 8, f(8) is 10. The difference is 16 - 10 = 6.
  10. -10 · fg(4): g(4) is 2, f(2) is 4. gf(4): f(4) is 16, g(16) is 14. The difference is 4 - 14 = -10.
Worksheet · LightMySky