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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 Calculate [3 marks]

    The diagram shows a function machine for \(f\). (a) Write down an expression for \(f(x)\). [1 mark] (b) Calculate \(f(3)\). [1 mark] (c) Solve \(f(x) = 11\). [1 mark]

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

    (a) \(f(x) = 5 - 2x\). (b) \(f(3) = 5 - 6 = -1\). (c) \(5 - 2x = 11\), so \(-2x = 6\) and \(x = -3\).

    Mark scheme

    • (a) \(-2x + 5\) or \(5 - 2x\) — B1
    • (b) \(-1\) — B1
    • (c) \(-3\) — B1
  2. 2 Calculate [4 marks]

    \(f(x) = x^2\) and \(g(x) = x + 2\) (a) Calculate \(fg(3)\). [1 mark] (b) Solve \(fg(x) = gf(x)\). [3 marks]

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    Model answer

    (a) \(g(3) = 5\), so \(fg(3) = f(5) = 25\). (b) \(fg(x) = (x + 2)^2\) and \(gf(x) = x^2 + 2\). So \((x + 2)^2 = x^2 + 2\), which gives \(x^2 + 4x + 4 = x^2 + 2\), so \(4x = -2\) and \(x = -\dfrac{1}{2}\).

    Mark scheme

    • (a) \(25\) — B1
    • (b) \(fg(x) = (x + 2)^2\) and \(gf(x) = x^2 + 2\) — M1
    • (b) \(x^2 + 4x + 4 = x^2 + 2\) — M1
    • (b) \(-\dfrac{1}{2}\) — A1
  3. 3 Find [3 marks]

    \(f(x) = \dfrac{x - 1}{4}\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Calculate \(f^{-1}(3)\). [1 mark]

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    Model answer

    (a) \(y = \dfrac{x - 1}{4}\), so \(4y = x - 1\) and \(x = 4y + 1\). So \(f^{-1}(x) = 4x + 1\). (b) \(f^{-1}(3) = 4 \times 3 + 1 = 13\).

    Mark scheme

    • (a) \(4y = x - 1\) or equivalent — M1
    • (a) \(f^{-1}(x) = 4x + 1\) — A1
    • (b) \(13\) — B1
  4. 4 Solve [4 marks]

    \(f(x) = 5x - 4\) (a) Find \(f^{-1}(x)\). [2 marks] (b) Solve \(f^{-1}(x) = f(x)\). [2 marks]

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    Model answer

    (a) \(f^{-1}(x) = \dfrac{x + 4}{5}\). (b) \(\dfrac{x + 4}{5} = 5x - 4\), so \(x + 4 = 25x - 20\), \(24 = 24x\) and \(x = 1\).

    Mark scheme

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

    \(f(x) = \dfrac{5}{x - 3}\) where \(x \ne 3\) (a) Find \(f^{-1}(x)\). [3 marks] (b) Explain why \(f^{-1}(0)\) cannot be calculated. [1 mark]

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    Model answer

    (a) \(y(x - 3) = 5\), so \(x - 3 = \dfrac{5}{y}\) and \(x = \dfrac{5}{y} + 3\). So \(f^{-1}(x) = \dfrac{5}{x} + 3\). (b) \(\dfrac{5}{0}\) is not defined, because you cannot divide by zero.

    Mark scheme

    • (a) \(y(x - 3) = 5\) — M1
    • (a) \(x = \dfrac{5}{y} + 3\) — M1
    • (a) \(f^{-1}(x) = \dfrac{5}{x} + 3\) — A1
    • (b) You cannot divide by zero — B1
  6. 6 Calculate [4 marks]

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

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    Model answer

    \(fg(x) = 4(ax + 3) - 1 = 4ax + 11\) and \(gf(x) = a(4x - 1) + 3 = 4ax - a + 3\). So \(11 = -a + 3\), which gives \(a = -8\).

    Mark scheme

    • \(fg(x) = 4ax + 11\) — M1
    • \(gf(x) = 4ax - a + 3\) — M1
    • \(11 = -a + 3\) — M1
    • \(-8\) — 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 \(a = -2\).