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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.

**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.

To solve equations or rearrange formulae we need to use **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.

A number machine is an alternative name for function machines. A number machine is a way of writing the rules which link the inputs and the outputs.

In order to solve an equation using a function machine:

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

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

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

DOWNLOAD FREEFunction 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:

Solve x + 5 = 12

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

The only operation is + \;5 .

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.**

Solve 4x=20

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

The only operation is \times\; 4 .

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

x = 20\div 4
x =5

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

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

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

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

x=\left( 3+6 \right)\times 2
x=18

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

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

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

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

x=\left( 45\div 5 \right)-2
x=7

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

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

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

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

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

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

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

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

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

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

**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.

E.g.

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

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

a=17,\;b=35

a=21,\;b=3

a=17,\;b=3

a=13,\;b=39

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.

m=-1,\; n=84

m=1,\; n=84

m=12, \;n=28

m=-12, \;n=28

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.

p=8, \;q=9

p=3, \;q=24

p=24, \;q=3

p=9, \;q=8

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

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

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

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

1. Here is a function machine

(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. ?

(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.

Fill in the missing numbers for the function machine.

**(2 Marks)**

Show answer

First box 2

**(1)**

Second box 7

**(1)**

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

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