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In this worksheet, students will learn to identify functions accurately and use function relationships to find missing inputs or outputs.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Language of Functions

Difficulty level:

### QUESTION 1 of 10

Functions are rules that link inputs and outputs.

In a function, each input has only one corresponding value as an output.

Here is an example of a function with a rule times by 2:

However, the picture below does not show a function, just a relationship between numbers:

This is because 1 is linked with two outputs, and in functions, each input must have only one corresponding output.

In this activity, we will identify functions accurately and use function relationships to find missing inputs or outputs.

Below are four representations of relationships:

Tick those in the list below which show functions.

A

B

C

D

A function links inputs and outputs with a rule: × 2 + 5.

Using this rule, what is the corresponding output if an input is -2?

A function links inputs and outputs with a rule: × 2 + 5.

Using this rule, what is the corresponding input if an output is 9?

What is the rule that links the inputs and outputs below?

Subtract 3

Multiply by 3

Divide by 3

What is the rule that links the inputs and outputs below?

Subtract 4

Multiply by 3

Match each function below to the rule that links its inputs and outputs.

 Add 5 Subtract 5 Multiply by 7 Divide by 2 A B C D

How many corresponding outputs can each of the inputs in a function be linked to?

1

2

7

An infinite number

The function below is missing two outputs.

What are they?

 Output: 6 Output: 7 Output: 10 Output: 11 Input: 3 Input: 5

A function has a rule of: &times; 3 + 1

What are the inputs which correspond to the outputs 13 and 19

 Input: 3 Input: 4 Input: 6 Input: 7 Output: 13 Output: 19

Match the inputs on the left with the outputs on the right for a function with a rule: multiply by 5, subtract 7.

## Column B

3
8
6
23
2
3
7
28
• Question 1

Below are four representations of relationships:

Tick those in the list below which show functions.

A
D
EDDIE SAYS
Only A and D show functions. Each element of the inputs in these examples is linked to exactly one element in the outputs. B is not a function because 11 is linked with both 3 and 4. C is not a function because 7 is not linked with any of the outputs. Does that make sense to you? Review the Introduction if you are all unsure before you move on to the rest of this activity.
• Question 2

A function links inputs and outputs with a rule: × 2 + 5.

Using this rule, what is the corresponding output if an input is -2?

1
EDDIE SAYS
Let's start with -2. Multiply -2 by 2 = -4 Now add 5 to -4 = 1 So, in this function, if the input is -2, then the output is 1.
• Question 3

A function links inputs and outputs with a rule: × 2 + 5.

Using this rule, what is the corresponding input if an output is 9?

2
EDDIE SAYS
Did you spot that we are asked for the input here rather than the output? To work out the input, we need to go backwards and use inverse operations. Instead of + 5, we need to - 5: 9 - 5 = 4 Now instead of multiplying by 2, we need to divide by 2: 4 ÷ 2 = 2 So, in this function, if the output is 9, then the input is 2.
• Question 4

What is the rule that links the inputs and outputs below?

Multiply by 3
EDDIE SAYS
Did you remember to go from inputs to outputs? To get from 2 to 6 we need to multiply by 3. From 7 to 21, we also multiply by 3. And from 5 to 15, we also multiply by 3. Can you spot any input and output pairs which do not fit this pattern? No, so × 3 is the rule of this function.
• Question 5

What is the rule that links the inputs and outputs below?

EDDIE SAYS
Think about how to get from 3 to 7, 12 to 16, or 9 to 13. Did you notice that in each case, we are +4 to reach the output? Do any pairs not fit this pattern? No, so + 4 is the rule of this function.
• Question 6

Match each function below to the rule that links its inputs and outputs.

 Add 5 Subtract 5 Multiply by 7 Divide by 2 A B C D
EDDIE SAYS
In each diagram, we need to identify the relationship between the inputs and outputs. In function A, we always multiply by 7 to convert the input into output ( e.g. 1 × 7 = 7; 5 × 7 = 35; etc.) In function B, the rule is to subtract 5 (e.g. 1 - 5 = -4; 4 - 5 = -1, etc.) In function C, we add 5 to each input in order to generate the output (e.g. 1 + 5 = 6; 4 + 5 = 9, etc.) Finally, in function D, each input is divided by 2 (8 %divide; 2 = 4; 24 ÷ 2 = 12; etc.) Did you match all those successfully?
• Question 7

How many corresponding outputs can each of the inputs in a function be linked to?

1
EDDIE SAYS
The definition of a function is that it links each input with only one output. If more than one input and output can be linked together, then the relationship shown is not a function. This is an important rule to remember.
• Question 8

The function below is missing two outputs.

What are they?

 Output: 6 Output: 7 Output: 10 Output: 11 Input: 3 Input: 5
EDDIE SAYS
Did you spot that each input is doubled first and then 1 is added? If we apply this rule to the inputs currently without outputs, we find that: 3 × 2 + 1 = 7 5 × 2 + 1 = 11 Did you match these inputs and outputs accurately in the table?
• Question 9

A function has a rule of: &times; 3 + 1

What are the inputs which correspond to the outputs 13 and 19

 Input: 3 Input: 4 Input: 6 Input: 7 Output: 13 Output: 19
EDDIE SAYS
To work out inputs, we need to go backwards from our outputs. So × 3 + 1 becomes -1 ÷3. So if our output is 13, our input will be: 13 - 1 ÷ 3 = 4 When our output is 19, our input will be: 19 - 1 ÷ 3 = 6 Did you match these inputs and outputs accurately in the table?
• Question 10

Match the inputs on the left with the outputs on the right for a function with a rule: multiply by 5, subtract 7.

## Column B

3
8
6
23
2
3
7
28
EDDIE SAYS
We need to multiply each number on the left by 3, before subtracting 7, from that answer. If we do this, we get the following calculations: 3 × 5 - 7 = 8 6 × 5 - 7 = 23 2 × 5 - 7 = 3 7 × 5 - 7 = 28 Great work identifying functions and finding missing inputs and outputs! Why not practise functions at a higher level if you are looking for a challenge and are feeling confident?
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