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GCSE Maths Algebra Functions

Function Machines

Function Machines

Here we will learn about function machines, including finding outputs, finding inputs and using function machines to solve equations.

There are also function machine worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are function machines?

Function machines are used to apply operations in a given order to a value known as the Input. The final value produced is known as the Output.

A function machine can be applied to numbers or be used for algebraic manipulation. They can be used to solve number problems, solve equations and rearrange formulae.

E.g.

Function Machines image 1 1

To solve equations or rearrange formulae we need to think about inverse operations and work backwards. We will see how to use function machines to solve equations on this page.

Not all equations can be solved using a function machine but they can be applied to a lot of situations where the unknown is on one side of the equation.

Function machines can be used to help produce tables of values for graphs such as quadratic or cubic graphs.

What are function machines?

What are function machines?

How to solve equations using a function machine

In order to solve an equation using a function machine:

  1. Consider the order of operations being applied to the unknown.
  2. Draw a function machine starting with the unknown as the Input and the value the equation is equal to as the Output.
  3. Work backwards, applying inverse operations to find the unknown Input.

Explain how to solve equations using a function machine

Explain how to solve equations using a function machine

Function machines worksheet

Function machines worksheet

Function machines worksheet

Get your free function machines worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Function machines worksheet

Function machines worksheet

Function machines worksheet

Get your free function machines worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Related functions lessons

Function machines is part of the larger topic, functions. You may find it useful to explore the main topic before looking into the detailed individual lessons below:

Solving equations using function machines examples

Example 1: solving a one-step-equation using a function machine

Solve x + 5 = 12

  1. Consider the order of operations being applied to the unknown.

The only operation is + \;5 .

2Draw a function machine starting with the unknown as the Input and the value the equation is equal to as the Output.

Function machines example 1 step2 1

3Work backwards, applying inverse operations to find the unknown Input.

x = 12 - 5

x = 7

Example 2: solving a one-step-equation using a function machine

Solve 4x=20

The only operation is \times\; 4 .

Function machines example 2 step 2 1

x = 20\div 4

x =5

Example 3: solving a two-step-equation using a function machine

Solve \frac{x}{2}-6=3

The order of operations is \div \;2, then - \;6 .

Function machines example 3 step 2 1

x=\left( 3+6 \right)\times 2

x=18

Example 4: solving a two-step-equation using a function machine

Solve 5\left( x+2 \right)=45

The order of operations is + \;2, then \times \; 5 .

Function machines example 3 step 2 2

x=\left( 45\div 5 \right)-2

x=7

Example 5: solving a two-step-equation using a function machine

Solve \frac{3p-1}{4}=8

The order of operations is \times\;3, then - \; 1 , then \div \; 4 .

Function machines example 5 step 2 1

\begin{aligned} & p=\left( 8\times 4+1 \right)\div 3 \\\\ & p=11 \end{aligned}

Example 6: solving a three-step-equation using a function machine

Solve \frac{2q+18}{3}=14

The order of operations is \times \; 2, then + \; 18 , then \div \; 3 .

Function machines example 6 step 2 1

\begin{aligned} & q=\left( 14\times 3-18 \right)\div 2 \\\\ & q=12 \end{aligned}

Common misconceptions

  • Not working backwards when using inverse functions

When using two-step function machines or others with more operations to solve equations, a common error is to forget to work backwards. The inverse operations are used but in the wrong order.

  • Not using the correct order of operations when drawing a function machine

A common error is to not follow the correct order of operations when creating a function machine for an equation.

For example, for the equation 2x-1=7 , the multiplication by two takes place before subtracting one.

Practice function machines questions

1. Find the missing Output and missing Input for the function machine.

 

Practice function machine questions 1 1

a=17,\;b=35
GCSE Quiz False

a=21,\;b=3
GCSE Quiz False

a=17,\;b=3
GCSE Quiz True

a=13,\;b=39
GCSE Quiz False

Work forwards to find a and backwards, using inverse operations, to find b.

2. Find the missing Output and missing Input for the function machine.

 

Practice function machine questions 2 1

m=-1,\; n=84
GCSE Quiz True

m=1,\; n=84
GCSE Quiz False

m=12, \;n=28
GCSE Quiz False

m=-12, \;n=28
GCSE Quiz False

Work forwards to find m and backwards, using inverse operations, to find n.

3. Find the missing Output and missing Input for the function machine.

 

Practice function machine questions 3 1

p=8, \;q=9
GCSE Quiz False

p=3, \;q=24
GCSE Quiz False

p=24, \;q=3
GCSE Quiz False

p=9, \;q=8
GCSE Quiz True

Work forwards to find p and backwards, using inverse operations, to find q.

4. Select the correct the function machine for the equation:

 

5x=10

Practice function machine questions 4 a

GCSE Quiz True

Practice function machine questions 4 b

GCSE Quiz False

Practice function machine questions 4 c

GCSE Quiz False

Practice function machine questions 4 d

GCSE Quiz False

x is the input, the operation is multiplying by 5 , the output is 10.

5. Select the correct the function machine for the equation:

 

2x-6=10

Practice function machine question 5a

GCSE Quiz False

Practice function machine question 5b

GCSE Quiz True

Practice function machine question 5c

GCSE Quiz False

Practice function machine question 5d

GCSE Quiz False

x is the input, the operation is multiplying by 2 , the second operation is subtracting 6, the output is 10.

6. Select the correct function machine and solution to the equation.

 

2x-4=18

Practice function machine questions 6 a

GCSE Quiz False

Practice function machine questions 6 b

GCSE Quiz False

Practice function machine questions 6 c

GCSE Quiz True

Practice function machine questions 6 d

GCSE Quiz False

Working backwards, you need to add four, and then divide by 2.

Function machines GCSE questions

1. Here is a function machine

 

Function machine GCSE questions 1 1

 

(a) What is the output when the input is 6 ?

 

(b) What is the output when the input is 10 ?

 

(2 Marks)

Show answer

(a) 40

(1)

 

(b) 68

(1)

2.  (a) Use the function machine to write a formula for y in terms of h. ?

 

Function machine GCSE questions 2a 1

 

(b) Use inverse operations to write a formula for h in terms of y.

 

(6 Marks)

Show answer

(a)

 

2h

(1)

 

2h+1

(1)

 

y=\frac{2h+1}{3}

(1)

 

(b)

 

3y

(1)

 

3y-1

(1)

 

h=\frac{3y-1}{2}

(1)

3.

Function machine GCSE questions 3 1

 

Fill in the missing numbers for the function machine.

 

(2 Marks)

Show answer

First box 2

(1)

 

Second box 7

(1)

Learning checklist

You have now learned how to:

  • Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships
  • Substitute values in expressions, rearrange and simplify expressions, and solve equations
  • Recognise and use relationships between operations including inverse operations
  • Where appropriate, interpret simple expressions as functions with inputs and outputs

The next lessons are

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