Function Machines
Function Machines is a key algebra topic at GCSE Maths. This Foundation worksheet gives you exam-style questions on using function machines, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Reverse the operations to find the input from a given output.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Challenge / Extension
Stretch yourself beyond the basics.
Topic overview
A function machine shows a sequence of operations applied to an input to produce an output. The input goes in one end, passes through each operation in order, and emerges as the output.
Reading them forwards is straightforward: apply each operation in turn. Reading backwards is the useful skill, because it shows how to reverse a process — and that is exactly how solving equations works.
To reverse a machine, start from the output and apply the inverse of each operation, working right to left. Add becomes subtract, multiply becomes divide, and the order is reversed as well as the operations.
Revision notes
Working forwards
Apply each operation to the input in the order shown.
An input of \(4\) into a machine that multiplies by \(3\) then adds \(5\) gives \(12\), then \(17\).
Working backwards
Start at the output and undo each operation, moving right to left.
With an output of \(17\), subtract \(5\) to get \(12\), then divide by \(3\) to get \(4\). Both the operations and their order are reversed.
Linking to algebra
A function machine is the same process as an expression. Multiplying by \(3\) then adding \(5\) is \(3x + 5\).
Solving \(3x + 5 = 17\) is exactly running the machine backwards, which is why the two topics reinforce each other.
Key points
- An input passes through operations to give an output.
- Apply operations left to right going forwards.
- Reverse both the operations and their order going backwards.
- Add becomes subtract; multiply becomes divide.
- A machine can be written as an algebraic expression.
- Reversing a machine is the same as solving an equation.
Worked examples
Example 1
An input of \(6\) is multiplied by \(4\) then decreased by \(9\). Find the output.
Working
Example 2
A machine adds \(7\) then multiplies by \(2\). The output is \(26\). Find the input.
Working
Example 3
Write the expression for a machine that multiplies by \(5\) then subtracts \(2\).
Working
Common mistakes
Undoing the operations in the original order.
Going backwards you must reverse the order as well as the operations.
Using the same operation instead of its inverse.
To undo an addition you subtract, not add again.
Applying operations in the wrong order forwards.
Follow the machine left to right exactly as drawn.
Writing the expression the wrong way round.
Multiply then subtract gives 5x − 2, not 5(x − 2).
Exam tips
- Draw the machine with arrows if the question gives only words.
- Write the inverse operations underneath before working backwards.
- Check by running your input forwards through the machine.
- Link the machine to an expression to connect it with equation solving.
Key terms
- Input
- The value entering the machine.
- Output
- The value produced by the machine.
- Inverse operation
- The operation that undoes another.
- Function
- A rule assigning one output to each input.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.