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Exam questions · Maths · Functions, Sequences and Rates of Change

Functions and Function Notation

  • 6 exam questions
  • 22 marks
  • 9 quick checks
  1. 1 Work out [3 marks]

    The diagram shows a function machine for \(f\). (a) Write down an expression for \(f(x)\). [1 mark] (b) Work out \(f(-2)\). [1 mark] (c) Solve \(f(x) = 15\). [1 mark]

    A function machine with two operations, used to work out the function f of x.
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    Model answer

    (a) \(f(x) = 2x + 7\). (b) \(f(-2) = 2 \times (-2) + 7 = 3\). (c) \(2x + 7 = 15\), so \(x = 4\).

    Mark scheme

    • (a) \(2x + 7\) — B1
    • (b) \(3\) — B1
    • (c) \(4\) — B1
  2. 2 Work out [4 marks]

    \(f(x) = 3x + 2\) and \(g(x) = x - 4\) (a) Work out \(fg(6)\). [2 marks] (b) Show that \(fg(x) = 3x - 10\). [2 marks]

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    Model answer

    (a) \(g(6) = 2\), then \(f(2) = 3 \times 2 + 2 = 8\). (b) \(fg(x) = f(x - 4) = 3(x - 4) + 2 = 3x - 12 + 2 = 3x - 10\).

    Mark scheme

    • (a) \(g(6) = 2\) — M1
    • (a) \(8\) — A1
    • (b) \(3(x - 4) + 2\) — M1
    • (b) \(3x - 12 + 2 = 3x - 10\), with the working shown — A1
  3. 3 Find [3 marks]

    \(f(x) = 5x - 2\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Work out \(f^{-1}(8)\). [1 mark]

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    Model answer

    (a) \(y = 5x - 2\), so \(x = \dfrac{y + 2}{5}\). So \(f^{-1}(x) = \dfrac{x + 2}{5}\). (b) \(f^{-1}(8) = \dfrac{10}{5} = 2\).

    Mark scheme

    • (a) \(5x = y + 2\) or equivalent — M1
    • (a) \(f^{-1}(x) = \dfrac{x + 2}{5}\) — A1
    • (b) \(2\) — B1
  4. 4 Solve [4 marks]

    \(f(x) = 3x - 1\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Solve \(f^{-1}(x) = f(x)\). [2 marks]

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    Model answer

    (a) \(f^{-1}(x) = \dfrac{x + 1}{3}\). (b) \(\dfrac{x + 1}{3} = 3x - 1\), so \(x + 1 = 9x - 3\), \(4 = 8x\) and \(x = \dfrac{1}{2}\).

    Mark scheme

    • (a) \(x = \dfrac{y + 1}{3}\) or equivalent — M1
    • (a) \(f^{-1}(x) = \dfrac{x + 1}{3}\) — A1
    • (b) \(\dfrac{x + 1}{3} = 3x - 1\) — M1
    • (b) \(\dfrac{1}{2}\) — A1
  5. 5 Find [4 marks]

    \(f(x) = \dfrac{x + 1}{x - 2}\) where \(x \ne 2\). Find \(f^{-1}(x)\). [4 marks]

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    Model answer

    \(y = \dfrac{x + 1}{x - 2}\), so \(y(x - 2) = x + 1\) and \(xy - 2y = x + 1\). Then \(xy - x = 2y + 1\), so \(x(y - 1) = 2y + 1\) and \(x = \dfrac{2y + 1}{y - 1}\). So \(f^{-1}(x) = \dfrac{2x + 1}{x - 1}\).

    Mark scheme

    • \(y(x - 2) = x + 1\) — M1
    • \(xy - 2y = x + 1\) — M1
    • \(x(y - 1) = 2y + 1\) — M1
    • \(f^{-1}(x) = \dfrac{2x + 1}{x - 1}\) — A1
  6. 6 Work out [4 marks]

    \(f(x) = 2x + 1\) and \(g(x) = ax - 3\), where \(a\) is a constant. \(fg(x) = gf(x)\). Work out the value of \(a\). [4 marks]

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    Model answer

    \(fg(x) = 2(ax - 3) + 1 = 2ax - 5\) and \(gf(x) = a(2x + 1) - 3 = 2ax + a - 3\). So \(-5 = a - 3\), which gives \(a = -2\).

    Mark scheme

    • \(fg(x) = 2ax - 5\) — M1
    • \(gf(x) = 2ax + a - 3\) — M1
    • \(-5 = a - 3\) — M1
    • \(-2\) — A1

Quick check

  1. 1

    \(f(x) = 3x - 5\). What is \(f(4)\)?

    1. A12
    2. B7
    3. C2
    4. D-5
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    B: 7

    \(3 \times 4 - 5 = 7\).

  2. 2

    \(f(x) = 2x + 1\). What is \(f(-3)\)?

    1. A\(-5\)
    2. B7
    3. C\(-6\)
    4. D5
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    A: \(-5\)

    \(2 \times (-3) + 1 = -5\).

  3. 3

    \(f(x) = x^2 + 1\) and \(g(x) = 2x\). What is \(fg(2)\)?

    1. A10
    2. B8
    3. C5
    4. D17
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    D: 17

    \(g(2) = 4\), then \(f(4) = 16 + 1 = 17\).

  4. 4

    \(f(x) = x + 3\) and \(g(x) = x^2\). What is \(gf(x)\)?

    1. A\(x^2 + 3\)
    2. B\(x^2 + 9\)
    3. C\((x + 3)^2\)
    4. D\(x + 9\)
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    C: \((x + 3)^2\)

    \(gf(x) = g(f(x)) = (x + 3)^2\).

  5. 5

    \(f(x) = 2x + 1\). What is \(ff(x)\)?

    1. A\(4x + 1\)
    2. B\(4x + 3\)
    3. C\(4x^2 + 1\)
    4. D\(2x + 2\)
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    B: \(4x + 3\)

    \(ff(x) = 2(2x + 1) + 1 = 4x + 3\).

  6. 6

    What is the inverse of \(f(x) = x + 7\)?

    1. A\(f^{-1}(x) = x - 7\)
    2. B\(f^{-1}(x) = \dfrac{1}{x + 7}\)
    3. C\(f^{-1}(x) = 7 - x\)
    4. D\(f^{-1}(x) = x + 7\)
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    A: \(f^{-1}(x) = x - 7\)

    The inverse reverses the rule, so you subtract 7.

  7. 7

    What is the inverse of \(f(x) = 3x - 5\)?

    1. A\(\dfrac{x - 5}{3}\)
    2. B\(\dfrac{x}{3} + 5\)
    3. C\(\dfrac{1}{3x - 5}\)
    4. D\(\dfrac{x + 5}{3}\)
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    D: \(\dfrac{x + 5}{3}\)

    \(y = 3x - 5\) gives \(x = \dfrac{y + 5}{3}\).

  8. 8

    \(f(x) = 5x - 4\). Solve \(f^{-1}(x) = f(x)\).

    1. A\(x = 0\)
    2. B\(x = -1\)
    3. C\(x = 1\)
    4. D\(x = 2\)
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    C: \(x = 1\)

    \(f^{-1}(x) = \dfrac{x + 4}{5}\), so \(\dfrac{x + 4}{5} = 5x - 4\), which gives \(24x = 24\).

  9. 9

    \(f(x) = 2x - 1\) and \(g(x) = ax + 3\), and \(fg(x) = gf(x)\). What is \(a\)?

    1. A\(a = 2\)
    2. B\(a = -2\)
    3. C\(a = 8\)
    4. D\(a = -8\)
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    B: \(a = -2\)

    \(fg(x) = 2(ax + 3) - 1 = 2ax + 5\) and \(gf(x) = a(2x - 1) + 3 = 2ax - a + 3\), so \(5 = 3 - a\) and \(a = -2\).