The function machine
Imagine a machine: you drop a number in (the input), the machine does something to it (the rule), and a new number comes out (the output).
Input 4 → × 3 → Output 12
This is called a function machine. Every input follows the same rule.
Input-output tables
An input-output table lists what goes in and what comes out:
| Input | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Output | 5 | 6 | 7 | 8 | 9 |
Rule: add 4.
Finding the rule
- Look for a pattern: compare each input with its output. Did it go up or down? By how much?
- Try a simple rule (add, subtract, multiply, divide).
- Test the rule on every pair.
| Input | Output | Input → output |
|---|---|---|
| 2 | 10 | × 5 |
| 3 | 15 | × 5 |
| 6 | 30 | × 5 |
The rule is × 5.
Two-step rules
Some machines do two things.
| Input | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Output | 3 | 5 | 7 | 9 |
- The outputs go up by 2, so try × 2: 2, 4, 6, 8.
- Each real output is 1 more: the rule is × 2, then + 1.
Working backwards
If you know the output and the rule, use the inverse operations in reverse order.
Rule: × 4, then − 3. Output: 25.
- Undo − 3: 25 + 3 = 28.
- Undo × 4: 28 ÷ 4 = 7.
- Input = 7.
Real-life input-output
| Input | Rule | Output |
|---|---|---|
| Number of tickets | × $12 | Total cost |
| Hours worked | × $15 | Pay |
| Number of cars | × 4 | Number of wheels |
Worked examples
Example 1: Rule: − 6. Input 20. Output?
20 − 6 = 14.
Example 2: Find the rule: inputs 3, 5, 8 → outputs 9, 15, 24.
Each output is 3 times the input: × 3.
Example 3: Rule: ÷ 2, then + 5. Output 12. Input?
12 − 5 = 7; 7 × 2 = 14.
Common mistakes
- Checking only one pair. 2 → 4 could be + 2 or × 2. Test more pairs.
- Undoing steps in the wrong order when working backwards.
- Reading the table the wrong way, from output to input.
Try it yourself
| Input | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Output | 7 | 10 | 13 | 16 |
- What is the rule?
- What is the output for input 10?
- Rule: × 6. Output 42. Input?
- Rule: + 9. Input 15. Output?
Answers: 1) × 3, then + 4 2) 34 3) 7 4) 24
Quick recap
- A function machine turns each input into an output using a rule.
- Find the rule by testing all pairs in the input-output table.
- Work backwards with inverse operations.