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Functions

Part of MathsExpressions and formulae

Key points

Two diagrams of one step function machines, one shown above the other. At the start of both function machines is the word input. At the end of both machines is the word output. The step in the top machine is multiply by three, with the step pointing to the right. The step in the bottom machine is divide by three, with the step pointing to the left. The divide by three step is highlighted in orange.
Function machines can be worked through from left to right to find output values from an input, or from right to left to find an input value from an output.

  • A relates an input to an output. One or more are applied to an input to give an output. There is one output for a given input.

  • A function may be represented:

    • using
    • as an (or an equation)
    • as a graph

  • An input value becomes an output value when the operations of a function machine are worked through from left to right.

  • An input value can be found from the output when the are worked through from right to left.

Understanding the correct order of operations (using BIDMAS) is useful when using function machines to write expressions.

Two diagrams of one step function machines, one shown above the other. At the start of both function machines is the word input. At the end of both machines is the word output. The step in the top machine is multiply by three, with the step pointing to the right. The step in the bottom machine is divide by three, with the step pointing to the left. The divide by three step is highlighted in orange.
Function machines can be worked through from left to right to find output values from an input, or from right to left to find an input value from an output.
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Using function machines

A single function machine represents a one-step function. To use a single function machine (input 鈫 output):

  1. Start with the input.

  2. Work through the function machine operation from left to right. The output is the result of the calculation.

A multi-step function can be presented using multiple function machines. To use a multi-step function machine (input 鈫 output):

  1. Start with the input.

  2. Work through each function machine operation for each step from left to right. The output of each step becomes the input for the next step to the right. The final output is the result of the whole calculation.

An input value can be found from the output value by working through the inverse operations of the function machine from right to left.

Examples

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  1. Example one. A diagram of a one step function machine. The input is five. The step in the machine is multiply by three. At the end of the machine is a question mark which is highlighted orange.

    A single function machine is used for one operation. An input is processed to give an output. The input for this function machine is 5. The output is unknown.

Question

Work out the input for the 3-step function machine.

A diagram of a three step function machine. The input is unknown and shown as a question mark highlighted in orange. The first step is add seven. The second step is multiply by three. The third step is subtract four. The output is twenty-nine.

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Using function machines to write expressions

  • When the input to a function machine is a , the output is an.

  • One function gives a single term output. The expression will have more terms when there are more steps in the function machine.

To write an expression from a function machine:

  1. Start with the variable.

  2. Work through the operation for each step from left to right.

  3. Write the expression using the correct for each process.

  4. Use brackets when an operation applies to more than one term.

Understanding notation used in algebra will help when writing expressions.

Examples

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  1. A diagram of a one step function machine. The input is x. The step in the machine is multiply by two. The output is two x.

    For a variable input, the output is an expression.

Question

Write the expression for this 3-step function machine.

A diagram of a three step function machine. The input is the variable n. The first step in the machine is divide by four. The second step is subtract one. The third step is multiply by six. The output is unknown and shown as a question mark highlighted in orange.

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Practise using function machines

Practise using function machines with this quiz. You may need a pen and paper to help you work out your answers.

Quiz

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Real-life maths

A satellite navigation system attached to the inside of a car windscreen, showing a planned route on a map.
A 'sat nav' identifies a sequence of directions (input) to direct you to your destination (output).

Functions are used in real life as a single-, or multi-step, set of processes. For example, turning the dial on a shower to a higher setting increases the volume or temperature of the water. Turning the dial is the input and the increased volume or temperature of the water are the output.

Functions are also used to process information. A 'sat nav' (satellite navigation system) identifies a sequence of directions to help you get to your destination. The order in which these directions are followed is important. The directions and steps of the journey are the input and the destination is the output.

A satellite navigation system attached to the inside of a car windscreen, showing a planned route on a map.
A 'sat nav' identifies a sequence of directions (input) to direct you to your destination (output).
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Game - Divided Islands

Using your maths skills, help to build bridges and bring light back to the islands in this free game from 麻豆官网首页入口 Bitesize.

Play the Divided Islands game!
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