Functions & Relationships: Flow Diagrams, Tables and Formulae

Input values, output values and rules — how the same relationship can be shown as a flow diagram, a table or a formula, and how to find a missing input, output or rule, with a clear diagram and a quiz.

A function is simply a rule that turns one number (the input) into another (the output). The same relationship can be written in several ways — and the exam will happily switch between them.

1Flow diagrams

A flow diagram: each input goes through the rule (×3 + 1) to produce its output. In algebra the same rule is y = 3x + 1.

Worked ExampleFind the outputs for the rule × 3 + 1 with inputs 1, 2, 3

1
Input 1: 1 × 3 = 3, then 3 + 1 = 4.
Apply the operations in the order shown.
2
Input 2: 2 × 3 + 1 = 7.
3
Input 3: 3 × 3 + 1 = 10.
Outputs: 4, 7, 10.

2The same rule as a table

A table shows the same information in rows: inputs on the top, outputs underneath. For the rule above the table reads 1→4, 2→7, 3→10, 4→13.

3The same rule as a formula

y = 3x + 1 (x is the input, y is the output)

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Flow diagram, table and formula are three descriptions of one relationship. Being able to move between them is exactly what the ATP calls "equivalent forms".

4Working backwards — finding the input

Worked ExampleFor the rule y = 3x + 1, find x when y = 22

1
Write the equation: 3x + 1 = 22.
Substitute the known output.
2
Subtract 1: 3x = 21.
Undo the +1 first.
3
Divide by 3: x = 7.
Check: 3(7) + 1 = 22 ✓

5Finding the rule from a table

Worked ExampleFind the rule: inputs 1, 2, 3, 4 give outputs 5, 8, 11, 14

1
Look at the jump between outputs: 5 → 8 → 11 → 14, always +3.
A constant difference of 3 means the rule contains × 3.
2
Test × 3: input 1 gives 3, but the output is 5 — that is 2 more.
Find the adjustment.
3
Rule: y = 3x + 2.
Check input 4: 3(4) + 2 = 14 ✓

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Rule: × 2 + 5. Find the output when the input is 6.
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6 × 2 + 5 = 17
2
Rule: y = 4x − 3. Find y when x = 5.
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20 − 3 = 17
3
Rule: y = 4x − 3. Find x when y = 17.
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4x = 20 → x = 5
4
Rule: × 10. Complete: 3 → ___ and ___ → 70
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3 → 30; 7 → 70
5
Find the rule: 1→3, 2→5, 3→7, 4→9
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Differences of 2 → y = 2x + 1
6
Find the rule: 1→6, 2→12, 3→18
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y = 6x
7
Rule: y = x² + 1. Find y when x = 4.
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16 + 1 = 17