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A-Level Maths: Core 3for Edexcel

C3.2 Algebra and functions 2

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Mappings

A mapping is made up of two sets and a rule that relates elements from the first set, the input set, to the elements of the second set, the output set (or image set).

Input setInput set Output setOutput setRule

The set of all permissible inputs is called the domain of the mapping.

The set of all corresponding outputs is called the range of the mapping.

For example, suppose we have the input set A = {1, 2, 3, 4, 5}.

This set represents the domain of the mapping.

DOMAIN RANGE

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Double and subtract 3

12345

12345DOMAIN

–11357

–11357 RANGE

A B

Mappings

This set is related to the output set B = {–1, 1, 3, 5, 7} by the rule “double and subtract 3”.

If we call this rule m we can write it using mapping notation as:

m: a → 2a – 3

Where a represents the elements in set A.

–11357

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1

4

9

–11

–22

–33

–11

–22

–33

Square

Types of mappings

A mapping can be described as:

One-to-one

In a one-to-one mapping, each element in the domain is mapped onto exactly one element in the range. For example:

Many-to-oneIn a many-to-one mapping, two or more elements in the domain can be mapped onto the same element in the range. For example:

1234

1234

–1012

–1012

Subtract 2

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5

6

7

8

1

2

4

8

1

2

4

8

Is a factor of

–11

–22

–33

–11

–22

–33

Square root

1

4

9

Types of mappings

One-to-many

In a one-to-many mapping each element in the domain can be mapped onto two or more elements in the range. For example:

Many-to-many

In a many-to-many mapping more than one element in the domain can be mapped onto more than one element in the range. For example:

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Functions

A function is a special type of mapping such that each member of the domain is mapped to one, and only one, element in the range.

In other words,

One-to-many and many-to-many mappings are not functions.

Most functions you will meet are defined over a continuous domain, such as the set of all real numbers.

Such functions are best represented by a graph plotting the elements in the domain along the horizontal axis against the corresponding members of the range along the vertical axis.

Only a one-to-one or a many-to-one mapping can be called a function.Only a one-to-one or a many-to-one mapping can be called a function.

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Which graphs represent functions?

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Function notation

We usually use the letter f to represent a function but other letters such as g and h can also be used.

The letter x is normally used to represent elements of the domain (input values).

For example, if we have the function “square and add 5” this can be written as

We can also use mapping notation to write

Since we can choose the value of x it is called an independent variable.

f(x) = x2 + 5

f: x → x2 + 5

f of x equals x squared plus 5

f is a function such that x is mapped onto

x squared plus 5.

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Function notation

The letter y is normally used to represent elements of the range (output values), so y = f(x).

Since the value of y is determined by the function acting on x it is called a dependent variable.

If we choose x to be –3, say, we can write

f(–3) = (–3)2 + 5 = 14

We say that 14 is the image of –3 under the function f.

Since the domain of f(x) = x2 + 5 has not been given, we assume that x belongs to the set of real numbers, .

What is the range of this function?

f(x) cannot be less than 5, so the range is:

f(x) ≥ 5

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Functions and mappings

Finding the range of a function

Composite functions

Inverse functions and their graphs

The modulus function

Transforming functions

Examination-style questions

Co

nte

nts

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Finding the range of a function

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The domain and range of a function

Remember,

A function is only fully defined if we are given both:

the rule that defines the function, for example f(x) = x – 4.

the domain of the function, for example the set {1, 2, 3, 4}.

Given the rule f(x) = x – 4 and the domain {1, 2, 3, 4} we can find the range:

{–3, –2, –1, 0}

The domain of a function is the set of values to which the function can be applied.

The range of a function is the set of all possible output values.

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The domain and range of a function

It is more common for a function to be defined over a continuous interval, rather than a set of discrete values. For example:

When x = –2, f(x) =

The range of the function is therefore

–8 – 7 = –15

When x = 5, f(x) = 20 – 7 = 13

Since this is a linear function, substitute the smallest and largest values of x:

–15 ≤ f(x) < 13

The function f(x) = 4x – 7 is defined over the domain –2 ≤ x < 5. Find the range of this function.