Input-output tables

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.

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.