Sheet № 201 · Foundation + Higher · AQA · Edexcel · OCR
Function Machines –
Function machines are one of the first introductions to algebraic thinking in GCSE Maths. They appear on Foundation and Higher papers and help bridge the gap between arithmetic and formal algebra.
§Key definitions
Question:
A function machine takes an input, multiplies by 4, then subtracts 3. Find the output when the input is 6.
Answer:
Input = 4
Q1 (Foundation):
A function machine multiplies by 5, then adds 2. Find the output when the input is 7.
Q2 (Foundation):
A function machine subtracts 4, then multiplies by 3. The output is 18. Find the input.
Q3 (Higher):
A function machine divides by 2, then squares the result. Write an expression for the output when the input is n.
§Formulas to memorise
To find the output, apply operations left to right in order
To find the input, apply inverse operations right to left (reverse order)
Worked example
A function machine takes an input, multiplies by 4, then subtracts 3. Find the output when the input is 6.
Working:
⚠ Common mistakes
- ✗Applying operations in the wrong order. Function machines go left to right. Multiplying by 2 then adding 3 gives a different result from adding 3 then multiplying by 2.
- ✗Using the wrong inverse operation. The inverse of adding is subtracting, and the inverse of multiplying is dividing. Students sometimes confuse these, especially with squaring and square-rooting.
- ✗Forgetting to reverse the order when working backwards. If the forward operations are multiply then add, the backward operations must be subtract then divide — in that order.
✦ Exam tips
- →Draw the function machine diagram if one is not given — it helps you see the order of operations.
- →When writing an expression, remember that "multiply by 3 then add 5" gives 3x + 5, not 3(x + 5).
- →Working backwards is the same skill as solving an equation — practise both to build confidence.