[discia] GCSE Maths › Algebra › Function Machines

Function Machines

discia.org 10 questions · 29 marks
Name Date
  1. Either paper 2 marks

    Here is a function machine.

    Input\(\times 5\)\(- 8\)Output
    \(3\)→→?

    Work out the output when the input is \(3\).

    Show answer
    \(7\)
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  2. Either paper 3 marks

    Here is a function machine.

    Input\(+ 2\)\(\times 6\)\(- 2\)Output
    \(3\)→→→?

    The input is \(3\).

    Work out the output.

    Show answer
    \(28\)
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  3. Either paper 3 marks

    Here is a function machine.

    Input\(\times 4\)\(- 7\)Output
    ?→→\(9\)

    The machine gives an output of \(9\).

    Find the input.

    Show answer
    \(4\)
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  4. Either paper 1 mark

    Anya wants to make a number machine so that \(y = 8x + 5\)

    Here is her attempt.

    Input \(x\)× 8− 5Output \(y\)
    \(x\)→→\(y\)

    What mistake has she made?

    Show answer
    Multiplied by 8 but subtracted 5 instead of adding it.
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  5. Either paper 4 marks

    Here is a function machine.

    Input\(- 4\)\(\times 6\)Output
    ?→→?
    (a)

    Calculate the output when the input is \(9\).

    (b)

    Calculate the input when the output is \(24\).

    Show answer
    (a) \(30\)
    (b) \(8\)
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  6. Either paper 2 marks

    Here is a function machine. The first operation is missing.

    Input\(\times \, ?\)\(- 7\)Output
    \(3\)→→\(14\)

    Work out the missing number.

    Show answer
    \(7\)
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  7. Either paper 4 marks

    Here are two number machines.

    Machine 1

    Input\(\times 7\)\(+ 5\)Output
    \(a\)→→?

    Machine 2

    Input\(\times 5\)\(+ 21\)Output
    \(a\)→→?

    The outputs are the same.

    Work out the value of \(a\).

    Show answer
    \(8\)
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  8. Either paper 3 marks

    Each number machine changes the input \(x\) into the output \(y\).

    (a)

    Complete this number machine so that \(y = 9x + 16\)

    Input\(?\)\(?\)Output
    \(x\)→→\(y\)
    (b)

    Complete this number machine so that \(y = 2x - 26\)

    Input\(?\)\(?\)Output
    \(x\)→→\(y\)
    (c)

    Complete this number machine so that \(y = x\)

    Input\(\div 9\)\( ?\)Output
    \(x\)→→\(y\)
    Show answer
    (a) The first box is \(\times 9\) and the second box is \(+ 16\)
    (b) The first box is \(\times 2\) and the second box is \(- 26\)
    (c) The second box is \(\times 9\)
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  9. Either paper 3 marks

    Each number machine changes an input into an output.

    (a)

    Here is a number machine.

    Input\(+ 10\)\(\times 6\)Output
    \(n\)→→\(d\)

    Write down the output \(d\) in terms of \(n\).

    (b)

    Here is another number machine, with both boxes left blank.

    Input\(?\)\(?\)Output
    \(u\)→→\(k\)

    Complete the number machine so that \(k = \frac{ u + 8 }{ 2 }\)

    Show answer
    (a) \(d = 6(n + 10)\)
    (b) The first box is \(+ 8\) and the second box is \(÷ 2\)
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  10. Either paper 4 marks

    Reuben is investigating number machines.

    (a)

    Here is a number machine.

    InputOutput
    \(x\)× 9+ 3\(y\)

    Show that when the input increases by 2, the output increases by 18.

    (b)

    Now \(f(x) = x^{ 2 }\)

    Reuben says that \(\frac{ f(6) }{ f(2) }\) is equal to \(f(3)\) because \(6 \div 2 = 3\).

    Is Reuben correct?

    Show working to support your answer.

    Show answer
    (a) Input \(x\) gives \(y = 9x + 3\). Input \(x + 2\) gives \(9(x + 2) + 3 = 9x + 18 + 3\). The difference is \((9x + 18 + 3) - (9x + 3) = 18\).
    (b) Yes. \(f(6) = 36\) and \(f(2) = 4\), so \(\frac{ f(6) }{ f(2) } = 9\), while \(f(3) = 9\). The ratio equals \(f(3)\).
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discia.org Function Machines · sheet 1
Answers Function Machines · sheet 1
  1. \(7\)
  2. \(28\)
  3. \(4\)
  4. Multiplied by 8 but subtracted 5 instead of adding it.
  5. (a) \(30\)
    (b) \(8\)
  6. \(7\)
  7. \(8\)
  8. (a) The first box is \(\times 9\) and the second box is \(+ 16\)
    (b) The first box is \(\times 2\) and the second box is \(- 26\)
    (c) The second box is \(\times 9\)
  9. (a) \(d = 6(n + 10)\)
    (b) The first box is \(+ 8\) and the second box is \(÷ 2\)
  10. (a) Input \(x\) gives \(y = 9x + 3\). Input \(x + 2\) gives \(9(x + 2) + 3 = 9x + 18 + 3\). The difference is \((9x + 18 + 3) - (9x + 3) = 18\).
    (b) Yes. \(f(6) = 36\) and \(f(2) = 4\), so \(\frac{ f(6) }{ f(2) } = 9\), while \(f(3) = 9\). The ratio equals \(f(3)\).
discia.org Function Machines · sheet 1