Function Machines and Simple Equations
A function machine applies one or more operations, such as 'x3, then +2', to an input number to produce an output. Working forwards finds the output from a given input; working backwards, using inverse operations in reverse order, finds the input from a given output, which is the same skill needed to solve a simple equation such as 3x + 2 = 17. 11+ papers test both directions, plus writing and solving simple equations directly.
Before you start
Make sure you're comfortable with these topics first:
Method
- For a function machine working forwards, apply each operation to the input in the order given, one step at a time.
- For a function machine working backwards, reverse both the order of the operations and each operation itself: + becomes -, x becomes divide, and vice versa.
- To solve a simple equation such as 3x + 2 = 17, treat it as a function machine in reverse: undo the +2 first by subtracting 2, then undo the x3 by dividing by 3.
- Always perform the same operation to both sides of an equation to keep it balanced.
- Check your answer by substituting it back into the original equation or function machine to confirm it gives the correct output.
- For two-step equations with brackets, such as 2(x + 3) = 16, divide both sides by the number outside the bracket first, then solve the simpler equation.
Worked example
A function machine has the rule 'multiply by 4, then subtract 5'. If the output is 23, find the input.
- Identify the two operations in order: x4, then -5.
- To work backwards, reverse the order and reverse each operation: first undo '-5' by adding 5, then undo 'x4' by dividing by 4.
- Add 5 to the output: 23 + 5 = 28.
- Divide by 4: 28 / 4 = 7.
- The input was 7. Check: 7 x 4 = 28, and 28 - 5 = 23, which matches the given output.
Practice questions
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Q1A function machine has the rule 'add 6, then multiply by 2'. Find the output for an input of 5.Show answer
Answer: 22 ((5 + 6) x 2)
Q2A function machine has the rule 'multiply by 3, then add 4'. If the output is 19, find the input.Show answer
Answer: 5 ((19 - 4) / 3)
Q3Solve: x + 9 = 17.Show answer
Answer: x = 8
Q4Solve: 5x = 35.Show answer
Answer: x = 7
Q5Solve: 2x - 3 = 11.Show answer
Answer: x = 7 (2x = 14, x = 7)
Q6Solve: 4(x + 1) = 24.Show answer
Answer: x = 5 (x + 1 = 6, x = 5)
Q7A function machine has the rule 'subtract 7, then divide by 3'. If the output is 4, find the input.Show answer
Answer: 19 (4 x 3 = 12, 12 + 7 = 19)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
A function machine has the rule 'multiply by 5, then subtract 8'. (a) Find the output for an input of 6. (b) If the output is 42, find the input.
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Solve the equation 3(x - 2) = 21, showing your method. Then check your answer by substituting it back into the original equation.
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Free printable worksheet
Want more practice on paper? Download the function machines and simple equations worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 19 of 11+ Maths Workbook 2, the whole course as one free printable PDF.
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