Think First
warm-up

Before diving into the practice questions, take a moment to think about what you already know about this topic.

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1
What You'll Master
objectives

Know

  • A function machine applies a rule to every input to produce an output

Understand

  • Reverse machines undo the rule — this is equivalent to solving an equation

Can Do

  • Follow a two-step rule through a function machine
  • Identify a rule from a table of input/output pairs
  • Use a reverse machine to find inputs from outputs
2
Words You Need
vocabulary
Function MachineA model that takes an input, applies a rule, and produces an output. Every input always gives exactly one output.
InputThe number you feed into a function machine. It is the starting value before the rule is applied.
OutputThe result you get after the machine applies its rule to the input.
RuleThe operation (or operations) a machine performs on every input. e.g. “multiply by 3, then add 2.”
Inverse MachineA reverse function machine that undoes the original rule using opposite operations in reverse order to find the input from a known output.
Two-Step RuleA rule that applies two operations in sequence — for example, “×3, then +2” gives the algebraic rule $y = 3x + 2$.
3
How Function Machines Work
+5 XP

A function machine takes an input, applies a rule step by step, and produces an output. The machine below uses the rule “multiply by 3, then add 2.”

INPUT 4 RULE STEP 1 ×3 → 12 RULE STEP 2 +2 → 14 OUTPUT 14

The same rule applied to all inputs produces a table of values:

Input ($x$) $\times 3$ $+2$ Output ($y$)
1355
2688
391111
4121414

The algebraic rule is $y = 3x + 2$.

4
Finding the Rule from a Table
+5 XP

Given a table of input/output pairs, you can work out the rule. If the output goes up by the same amount each time the input goes up by 1, it's a linear rule of the form $y = mx + c$.

Method:

  1. Find the constant difference in the outputs — this is the gradient $m$.
  2. Substitute one input/output pair into $y = mx + c$ to find $c$.
  3. Write the complete rule and verify with a second pair.
Input Output 1 7 2 11 3 15 4 19 +4 +4 +4
$y = 4x + 3$

Full working for the example (1,7), (2,11), (3,15), (4,19):

  • Constant difference in outputs: $11-7 = 4$, so $m = 4$
  • Substitute $(1,\,7)$: $7 = 4(1) + c \;\Rightarrow\; c = 3$
  • Rule: $y = 4x + 3$
  • Check with $(2,\,11)$: $4(2)+3 = 11$ ✓
Difference = gradient
The constant difference between outputs (when inputs go up by 1) IS the $m$ in $y = mx + c$.
Find $c$ by substituting
Once you have $m$, pick any row from the table and solve for $c$.
Always check twice
Verify your rule with at least two different input/output pairs before committing.
5
Reverse Function Machines (Solving Equations)
+5 XP

To find the input when you know the output, run the machine backwards using inverse operations in reverse order.

FORWARD (find output) In: 4 ×3 = 12 +2 = 14 Out: 14 REVERSE (find input) In: 14 −2 = 12 ÷3 = 4 Out: 4

The key insight: running a reverse machine is the same as solving an equation.

Machine rule “×3 then +2, output = 14” is identical to solving $3x + 2 = 14$:

  • $3x + 2 = 14$
  • Subtract 2: $3x = 12$
  • Divide by 3: $x = 4$
Reverse order
Undo the last operation first. If the machine did ×3 then +2, undo −2 first, then ÷3.
Inverse operations
$+$ undoes $-$  •  $-$ undoes $+$  •  $\times$ undoes $\div$  •  $\div$ undoes $\times$.
It's equation-solving
Every reverse machine problem is really an algebra equation in disguise.

Watch Me Solve It · Worked example

Watch Me Solve It · Reverse machine
+15 XP per step
Q
PROBLEM
A machine rule is “×5 then −3.” The output is 22. Find the input.
  1. 1
    Write the reverse sequence
    Output 22 → undo −3 (add 3) → undo ×5 (divide by 5) → Input
    Because the machine last did −3, we undo that first. Then we undo ×5 by dividing.
  2. 2
    Apply the reverse operations
    $22 + 3 = 25 \qquad 25 \div 5 = 5$
    Add 3 to undo the −3, giving 25. Then divide by 5 to undo the ×5, giving 5.
  3. 3
    Check by running forwards
    $5 \times 5 - 3 = 25 - 3 = 22$ ✓
    Substitute the input back into the original rule to confirm the output is 22.
Answer Input = 5  •  Check: $5 \times 5 - 3 = 22$ ✓
6
Common Pitfalls
heads-up
Checking only one input/output pair
If you only test one pair to verify your rule, a wrong rule can appear to work by coincidence. Always check at least two pairs.
Fix: substitute at least two different input/output pairs from the table before declaring the rule correct.
Undoing operations in the wrong order
Reverse machines undo in REVERSE order — not the same order as the forward machine. If forward is “×3 then +2”, reverse is “−2 then ÷3”, not “÷3 then −2”.
Fix: list the forward steps top to bottom, then read the inverse operations from bottom to top.
Confusing $y = 4x+3$ with $y = 4(x+3)$
$y = 4x + 3$ means “multiply $x$ by 4, then add 3.” But $y = 4(x+3)$ means “add 3 to $x$ first, then multiply by 4.” These give different outputs for the same input.
Fix: always check the order of operations. Brackets force an operation to happen first.

Quick Check · 5 questions

1
A function machine doubles the input, then subtracts 3. What is the output when the input is 6?
+10 XP
2
What is the rule for this function machine?
+10 XP
3
A function machine multiplies by 3, then adds 4. If the output is 25, what was the input?
+10 XP
4
Which function machine produces the sequence 6, 10, 14, 18, 22, ...?
+10 XP
5
A function machine: input $x$ → subtract 3 → multiply by 5 → output. What is the algebraic rule?
+10 XP
Apply Easy 2 MARKS

Q1. Complete this function machine table. Rule: multiply by 2, then add 7.

Answer in your workbook.
Apply Medium 3 MARKS

Q2. Find the rule for this function machine:

Answer in your workbook.
Apply Hard 4 MARKS

Q3. A function machine: input → add 4 → multiply by 3 → output. If the output is 33, find the input. Show your working.

Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins

Extension Problems

Ready for a bigger challenge? Try these extension problems.

R
Quick Review

Key Concept

Review the main ideas from this lesson.

Formulas

Key formulas and rules.

Watch Out

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Check

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Interactive: Algebra Machine

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Brain Trainer

⚡
Speed Drills — Find the Rule!

Set a timer for 3 minutes. Find the rule for each input-output table.

In: 1,2,3,4 → Out: 5,8,11,14
In: 1,2,3,4 → Out: 6,11,16,21
In: 1,2,3,4 → Out: 3,5,7,9
In: 1,2,3,4 → Out: 9,13,17,21
In: 1,2,3,4 → Out: 1,4,7,10
In: 1,2,3,4 → Out: 8,14,20,26