# Function Machines

## Function Machines Revision

**Function Machines**

A **function machine** takes an input, applies a series of mathematical operations to it, and then outputs a result. **Function machines** are quite a common question type, especially in foundation maths papers. There are **3** main types of questions you could be asked.

Make sure you are happy with the following topics before continuing.

**Type 1: Input to Output**

Here, you will be given an input and a function machine and then asked to produce the output.

**Example:** A function machine is shown below, what is the output if the input is 10?

**Type 2: Output Back to Input**

Here, you will be given an Output, a function machine, and then asked to find the Input which produced it.

**Example:** A function machine is shown below. What was the input if the output is 48?

If the output is 48 we need to follow the operations of the function machine in reverse and do the opposite operation.

First we must reverse **O****peration 2 **which is \times 3, this means we must \div \, 3.

48 \div 3 = 16

Next, **Operation 1 **which is -10, this means we must +10.

16+10 = 26

This shows the input was 26

**Type 3: Creating Function Machines**

You may be asked to create a function machine from an equation given.

**Example:** Draw a function machine below so that b = \dfrac{a-3}{5}

Here we have an input of a and an output of b.

We must next look at the order of operations, for this we will use BIDMAS

As this is a fraction the first operation will be a-3

The Second would then be (a-3) \div 5

Using this we can create the function machine shown below.

**Example: Function Machines Problems**

Inputting 5 into the **function machine** below would give an output of 4.

Find the value of x in the function machine below.

**[2 marks]**

So, if we input 5 into the function machine then we must first add \textcolor{orange}3:

5+3=8

Now, the question tells us that the result of dividing 8 by some number gives us 4, so we divide by \textcolor{orange}2 to complete the **function machine**

x = 2

## Function Machines Example Questions

**Question 1:** Below is a function machine,

a) Work out the output produced by an input of 35.

**[1 mark]**

b) Work out the input needed to produce an output of 48.

**[1 mark]**

a) Inputting 35, we first multiply by 3:

35 \times 3 = 105

Then, add 15 to get

105+15=120

b) We must work backwards and do the opposite operations. So, first subtracting 15 from the given output, we get

48-15=33

Then, dividing by 3 we get,

33\div 3=11

Meaning that 11 is the input required to give an output of 48.

**Question 2:** Below is a function machine.

Work out the output produced by an input of -5

**[1 mark]**

Inputting -5, we first multiply by -2:

-5\times-2=10

Then, adding 7 we get,

10+7=17

Meaning that -5 is the input required to give an output of 17.

**Question 3:** Below is a function machine.

Work out the output produced by an input of 3x

**[1 mark]**

Inputting 3x, we first multiply by \dfrac{1}{2}

3x\times\dfrac{1}{2}=\dfrac{3}{2}x

Then, dividing by 3 we get,

\dfrac{3}{2}x\div3=\dfrac{1}{2}x

Meaning that 3x is the input required to give an output of \dfrac{1}{2}x

**Question 4:** Below is a function machine. Give two operations that would output 27 from an input of 9

**[2 marks]**

Two operations are,

Multiplying by 4, so 9 \times 4 = 36

Subtracting 9, so 36 - 9 = 27

Any combination of operations which produce an output of 27 would be a correct answer.

**Question 5:** Below is a function machine.

When x is input into the function machine, 2x is the output. Find the value of x.

**[2 marks]**

We have to see what we get if we input x into the function machine. First, multiplying x by 12,

12\times x=12x

Then, subtracting 25 to get

12x-25

This is the output of inputting x but we know the output is equal to 2x, so we are left with the equation

12x-25=2x

Now we can solve this equation for x. Subtracting 2x from both sides, we get

10x-25=0

Adding 25 to both sides, we get

10x=25

Finally, dividing both sides by 10 we find x to be

x=25\div 10=2.5