Functions - KS3 Maths - BBC Bitesize

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. — Image caption,
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.

Using function machines

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

Start with the input.
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 → output):

Start with the input.
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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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.
Image caption,
To find the output, start with the input (5). Process the function machine operation. 5 is multiplied by 3. The output is the result of the calculation. 5 × 3 = 15. The output is 15
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A multi-step function machine may be used for more than one operation. To find the output, the operations are worked through from left to right.
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The input is 5 and the output is unknown. The 2-step function machine shows two operations. The first step is to multiply by 3. The second step is to add 2
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To find the output, start with the input (5). Work through the first step. 5 × 3 = 15. The output (15) becomes the input for the next step to the right.
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The output of the previous step (15) is the input for the second step. Work through the second calculation (15 + 2). The output is the result of all the working out. 3 × 5 = 15 and 15 + 2 = 17. The output is 17
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Function machines may be used in reverse order. Work through the inverse operations from a given output to find the original input. The inverse of multiplication is division.
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The output is 15. The input is unknown.
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Write the inverse function machine so that the direction goes from right to left. Write the inverse operation in the function machine. The inverse of multiply by 3 is divide by 3
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Start with the output (15). Work through the inverse operation (15 ÷ 3). The input is the result of the inverse calculation. 15 ÷ 3 = 5. The input is 5

Question

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

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:

Start with the variable.
Work through the operation for each step from left to right.
Write the expression using the correct for each process.
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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For a variable input, the output is an expression.
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For each function machine, the output is an algebraic expression.
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Start with the variable (𝒙 and 𝒚). Work through the operation for each function machine from left to right using the correct algebraic notation for each process. 𝒙 multiplied by 2 is 2𝒙. 𝒚 plus 5 is 𝒚 + 5
Image caption,
Write the expression for this 2-step function machine.
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Start with the variable (𝒎). Work through the operation for each step from left to right. 𝒎 multiplied by 2 is 2𝒎. Adding 5 gives the expression 2𝒎 + 5
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Write the expression for this 2-step function machine.
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Start with the variable (𝒏). Work through the operation for each step in the function machine from left to right. 𝒏 plus 5 is 𝒏 + 5. The whole of 𝒏 + 5 is multiplied by 2 and brackets are used to show this. Write the expression with the correct notation. The expression is 2(𝒏 + 5).
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For an input of 𝒏, a 2-step function machine gives the output 𝒏 − 3, all over 5. Write the correct operations to complete the steps for the function machine.
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The first operation that has been used on 𝒏 is subtract 3. This is 𝒏 − 3. The first step is − 3
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The next part of the expression is a division. 5 is the denominator of a fraction, so the operation is divide by 5. The second step for the function machine is ÷ 5

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.

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

Real-life maths

A satellite navigation system attached to the inside of a car windscreen, showing a planned route on a map. — Image caption,
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.