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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 \(f(4)\). (1 mark) (c) Solve \(f(x) = 16\). (1 mark)

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

    (a) \(f(x) = 3x - 5\). (b) \(f(4) = 3 \times 4 - 5 = 7\). (c) \(3x - 5 = 16\), so \(x = 7\).

    Mark scheme

    • (a) \(3x - 5\) — B1
    • (b) \(7\) — B1
    • (c) \(7\) — B1
  2. 2 Work out [4 marks]

    \(f(x) = 2x - 1\) and \(g(x) = x^2\) (a) Work out \(fg(3)\). (2 marks) (b) Find \(gf(x)\), giving your answer in the form \(ax^2 + bx + c\). (2 marks)

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    Model answer

    (a) \(g(3) = 9\), then \(f(9) = 2 \times 9 - 1 = 17\). (b) \(gf(x) = (2x - 1)^2 = 4x^2 - 4x + 1\).

    Mark scheme

    • (a) \(g(3) = 9\) — M1
    • (a) \(17\) — A1
    • (b) \((2x - 1)^2\) — M1
    • (b) \(4x^2 - 4x + 1\) — A1
  3. 3 Find [3 marks]

    \(f(x) = \dfrac{x + 4}{3}\) (a) Find \(f^{-1}(x)\). (2 marks) (b) Work out \(f^{-1}(5)\). (1 mark)

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    Model answer

    (a) \(y = \dfrac{x + 4}{3}\), so \(3y = x + 4\) and \(x = 3y - 4\). So \(f^{-1}(x) = 3x - 4\). (b) \(f^{-1}(5) = 3 \times 5 - 4 = 11\).

    Mark scheme

    • (a) \(3y = x + 4\) or equivalent — M1
    • (a) \(f^{-1}(x) = 3x - 4\) — A1
    • (b) \(11\) — B1
  4. 4 Solve [4 marks]

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

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    Model answer

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

    Mark scheme

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

    \(f(x) = \dfrac{3}{x + 2}\) where \(x \ne -2\) (a) Find \(f^{-1}(x)\). (3 marks) (b) Explain why \(f^{-1}(0)\) cannot be worked out. (1 mark)

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    Model answer

    (a) \(y(x + 2) = 3\), so \(x + 2 = \dfrac{3}{y}\) and \(x = \dfrac{3}{y} - 2\). So \(f^{-1}(x) = \dfrac{3}{x} - 2\). (b) \(\dfrac{3}{0}\) is not defined, because you cannot divide by zero.

    Mark scheme

    • (a) \(y(x + 2) = 3\) — M1
    • (a) \(x = \dfrac{3}{y} - 2\) — M1
    • (a) \(f^{-1}(x) = \dfrac{3}{x} - 2\) — A1
    • (b) You cannot divide by zero — C1
  6. 6 Work out [4 marks]

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

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    Model answer

    \(fg(x) = 3(ax - 4) + 2 = 3ax - 10\) and \(gf(x) = a(3x + 2) - 4 = 3ax + 2a - 4\). So \(-10 = 2a - 4\), which gives \(a = -3\).

    Mark scheme

    • \(fg(x) = 3ax - 10\) — M1
    • \(gf(x) = 3ax + 2a - 4\) — M1
    • \(-10 = 2a - 4\) — M1
    • \(-3\) — 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 \(a = -2\).