Function Machines and Inverse Operations
A function machine applies one or more operations, in order, to an input number to produce an output number. Inverse operations undo an operation, addition undoing subtraction, multiplication undoing division and squaring undoing square rooting, and running a function machine backwards using inverse operations in reverse order is the skill that leads directly into solving equations.
Before you start
Make sure you're comfortable with these topics first:
Method
- Read the function machine, or the written rule, to identify each operation applied to the input, in the order they are applied.
- To find an output from a given input, apply each operation in order, working left to right.
- To find an input from a given output, work backwards through the machine using the inverse of each operation.
- Apply the inverse operations in the REVERSE order to how the original operations were applied, undoing the last operation first.
- Remember the inverse pairs: addition undoes subtraction, multiplication undoes division, and squaring undoes square rooting.
- Check the answer by putting it back through the original function machine in the forward direction to see that it gives the stated output.
Worked example
A function machine takes an input, multiplies it by 3, then adds 4, to give an output of 25. Find the input.
- The forward machine is: input, then multiply by 3, then add 4, to give the output.
- To reverse it, undo the operations in reverse order, starting from the output of 25.
- Undo 'add 4' first by subtracting 4: 25 - 4 = 21.
- Undo 'multiply by 3' by dividing by 3: 21 / 3 = 7.
- Check: 7 x 3 = 21, then 21 + 4 = 25, which matches the given output, so the input is 7.
Practice questions
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Q1A function machine multiplies the input by 2, then adds 5. Find the output when the input is 6.Show answer
Answer: 17
Q2A function machine subtracts 3 from the input, then multiplies by 4. Find the output when the input is 9.Show answer
Answer: 24
Q3A function machine adds 7 to the input to give an output of 15. Find the input.Show answer
Answer: 8
Q4A function machine multiplies the input by 5 to give an output of 40. Find the input.Show answer
Answer: 8
Q5A function machine doubles the input, then subtracts 3, to give an output of 11. Find the input.Show answer
Answer: 7
Q6A function machine divides the input by 2, then adds 6, to give an output of 10. Find the input.Show answer
Answer: 8
Q7State the inverse operation of 'subtract 9'.Show answer
Answer: Add 9
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A function machine takes an input, subtracts 5, then multiplies by 2, to give an output of 18. Find the input.
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A function machine adds 4 to the input, then divides the result by 3, to give an output of 6. Find the input.
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Free printable worksheet
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This topic is chapter 22 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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