## What you need to know

### 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. Having a good knowledge of the following topics helps

### Example 1: Function Machines

Using the function machine below complete the questions

a)Work out the output we get when we input 7.

b)Work out the input needed to give 11 as an output.

c)Work out the output we get when we input 2x.

a) As we can see, the function machine applies two operations: “\times 4” and “+1”, specifically in that order.

So, with 7 as an input, first times by 4:

7\times 4=28

Then, subtract 1 to get the answer to be

28-1=27

b) To get an input from an output, we have to work backwards, doing the opposite operations in the opposite order.

Working backwards, we can see that the last operation is to subtract 1, so we must add 1 to our given output:

11+1=12

Then divide by 4 to give

12 \div 4 = 3

CHECK: you can check your answer by putting 3 through the function machine and seeing if the output matches 11.

c) This time, we have to put some algebra into the machine, but the principle is example the same.

Our first operation is multiplying by 4, so we get

4\times 2x=8x

Then, subtracting 1 we get

8x-1

You can see that we get an expression as an answer, not a number. This is to be expected any time you have to feed algebra through a function machine.

### Example 2: Function Machines

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

a)Fill in the missing number in the function machine.

b)Using the function, work out the output produced by an input of -6

a) So, if we input 5 into the function machine then we must first add 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 2 to complete the function machine

b) Inputting -6, we must first add 3.

-6+3=-3

Now we know the second step is to divide by 2, so we get

-3\div 2=-1.5

### Example Questions

1) Below is a function machine,

\text{Input}\longrightarrow \boxed{ \times3 } \longrightarrow \boxed{ +15 } \longrightarrow \text{Output}

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

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

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

2) Below is a function machine,

\text{Input}\longrightarrow \boxed{ \times-2 } \longrightarrow \boxed{ +7 } \longrightarrow \text{Output}

Work out the output produced by an input of -5

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

3) Below is a function machine,

\text{Input}\longrightarrow \boxed{ \times\dfrac{1}{2} } \longrightarrow \boxed{ \div 3 } \longrightarrow \text{Output}

Work out the output produced by an input of 3x

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

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

9 \longrightarrow \boxed{ ? } \longrightarrow 27

Two examples are,

9\times 3=27, so one operation would be multiplying by 3.

9+18=27, so a second operation would be adding 18.

5) Below is a function machine.

\text{Input}\longrightarrow \boxed{ \times12 } \longrightarrow \boxed{ -25 } \longrightarrow \text{Output}

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

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

### Worksheets and Exam Questions

#### (NEW) Function Machines Exam Style Questions - MME

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