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7. Algorithms
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Australia
Year 4

7.02 Functions Machines

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Introduction

A sausage

If you think of a sausage machine: something goes in, the machine works its magic, and sausages come out the other end.

A function machine is a little like that, just with no sausages. We have three important things that happen though, just like a sausage machine.

Function machines

With a function machine, we have:

  • something going in (input)

  • something happening in the machine (operation)

  • something coming out (output)

Is that all? Yes. It looks just like this.

An arrow with the word input pointing to a box with the word operation, then an arrow with the word output pointing outward.

Let's watch a video to see how it works and how it can go in reverse as well.

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    Examples

    Example 1

    Look at this function machine.

    A function machine with an input of 2, operation of plus 1, and missing output.
    a

    In the case above, what is the output of the function machine?

    Worked Solution
    Create a strategy

    Apply the operation to the input.

    Apply the idea

    The operation tells us to add 11 to the input.

    Output\displaystyle \text{Output}=\displaystyle =2+1\displaystyle 2+1Add 11 to the input
    =\displaystyle =3\displaystyle 3Evaluate
    b

    If the input was changed to 77, what would the new output of the function machine be?

    Worked Solution
    Create a strategy

    Apply the operation to the new input.

    Apply the idea

    The operation tells us to add 11 to the new input.

    A function machine with an input of 7, operation of plus 1, and a missing output.
    New output\displaystyle \text{New output}=\displaystyle =7+1\displaystyle 7+1Add 11 to the new input
    =\displaystyle =8\displaystyle 8Evaluate

    Example 2

    Look at this function machine.

    A function machine with an input of 5, a missing operation, and an output of 7.
    a

    What is the operation of this function machine?

    Worked Solution
    Create a strategy

    Think about what operation needs to happen to the input to get the output.

    Apply the idea

    To get from 55 to 77 we need to add 2.2.

    So, the operation is +2+2.

    b

    We can reverse the direction of a function machine.

    A function machine in reverse direction with an input of 7, a missing operation, and an output of 5.

    Now 77 is the input and 55 is the output.

    What is the operation of this function machine now?

    Worked Solution
    Create a strategy

    Reverse the mathematical operation found in part (a).

    Apply the idea

    We knew from part (a) that the operation is +2+2, which has a reverse operation of −2-2.

    Example 3

    Look at this function machine.

    A function machine with missing input, operation of plus 5, and an output of 12.

    In the case above, what is the input of the function machine?

    Worked Solution
    Create a strategy

    Reverse the operation and apply it to the output.

    Apply the idea

    The reverse of +5+5 is −5-5, which we are going to apply to the output.

    Input\displaystyle \text{Input}=\displaystyle =12−5\displaystyle 12-5Subtract 55 from the output
    =\displaystyle =7\displaystyle 7Evaluate
    Idea summary

    A function machine has an input to which an operation is applied and then gives an output:

    An arrow with the word input pointing to a box with the word operation, then an arrow with the word output pointing outward.
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