Definition OF Mapping and Functions

 

WEEK EIGHT

SS1 FURTHER MATHS FIRST TERM

MAPPING AND FUNCTIONS

CONTENT

Definition OF Mapping and Functions

Types of Mapping and function

SUB TOPIC: DEFINITION AND PROPERTIES OF MAPPING AND FUNCTION

A mapping is a rule which assigns every element in a set A to a distinct element in a set B, given that A and B are two non- empty sets. In set notation form, the rule which assigns an element ϵ A a unique element ϵ B, is called a mapping. The set A is called the Domain of the mapping while the set B is called the Co-domain.

The operative rule in all such assignment must be that “no child can have two fathers”, meaning, no one elements in the domain should be mapped to two elements in the co-domain.

If the rule which associates each element ϵ A, a unique element ϵ B is denoted by the mapping between the set A and the set B can be represented by any of the following notations:

A B or :A–––>B or

Where is the unique element in B which corresponds to the element in A.

A Subset of the co-domain, which is a collection of all the images of the elements of the domain, is called the range.

A mapping whose co-domain is a set of numbers is called a function.

Example 1:

Given that P and Q. Let the mapping be defined by the arrow diagram below:

 

 

b

f

e

g

f

 

As is the image of both is the image of while is the image of under the mapping. We observe that although every element of P has a unique image in Q, not all the elements of Q are images of P. This is one property of mapping.

The set is called the domain of the mapping; the set is the co-domain, while the set is the range. If we denote the set then.

Note the following about mapping: (i) it can be represented by any of the following graphical form; arrow diagrams, parallel axes, and Cartesian graph. (ii) A mapping may sometimes be regarded as a set of all ordered pairs .

Example 2:

Consider the relation shown in the arrow diagram shown below:

X f Y

p

q

r

e

f

g

The relation is not a mapping, not because g is not an image of any element in but because has no image in .

A function of a variable is a rule that describes how a value of the variable is manipulated to generate a value of the variable. The rule is often expressed in the form of an equation with a condition attached that for any input there is a unique value for. Consider the mapping (function) below;

 

 

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Let’s discover the rule which associates an element from a unique image in the set Y. It appears that when 1 is added to twice an element in X, it produces the corresponding image in the set Y.

Hence, if then we can write this as

.

Note that, though functions are rules, not all rules are functions. Consider the equation which is the same as

The rule ‘take the positive and negative square roots of the value ’ is a rule that is not a function because to each value of the input there are two different values of output .

Example 3:

Which of the following equations express rules that are functions?

Solution :

(a) does not express a function because to each value of there are two values of , positive and negative because . Indeed, any even root produces two variables.

(b) does express a function because to each value of there is just one value of .

CLASS ACTIVITIES:

1. Let . The arrow diagram below illustrates a mapping

 

 

-3

0

3

1

10

 

 

 

1. Determine the rule of the mapping. (b) What is the range of the mapping?
2. which of the following equations expresses a rule that is a function: (a) (b) (c)
3. Given the two functions expressed by :

, find the domain and range of: (a) (b)

SUB TOPIC: TYPES OF MAPPING AND FUNCTION

1. One – One – Mapping: This is a mapping where different elements in the domain have different images in the co-domain. Thus given a mapping if implies that implies that then the mapping is called a One-One mapping.

Examples of One – One mapping includes; the mapping which associates each state in Nigeria with its Governor, the mapping which assigns each university in west Africa with its Vice-Chancellor and the mapping which assigns each DLHS campus in Nigeria with its principal.

1. Onto Mapping: Given a mapping , if every element of the co-domain is an image of at least one element in the domain, then the mapping is called an Onto mapping. This implies that the Range of an Onto mapping is equal to the co-domain.

Example: Let the mapping be defined by the arrow diagram below;

 

A

 

-1

0

1

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9

0

1

 

The range of the mapping is equal to co-domain, hence the mapping is Onto. It is however not One-One because have the same image i.e. 1.

Example 2: Let be defined by , where is the set of real numbers. Find the domain and range of and state whether is One – One or Onto.

Solution: The domain of are all the set of real numbers, while the range are real numbers ≥ 2 because given = negative integer, will give a positive solution > 2 and will make to be equal to 2. is neither One – One nor Onto. It is not Onto because the range of is not equal to the co-domain as it excludes negative integers, 0 and numbers 1. It is not One – One the negative and positive value of a given digit e.g. -2 and 2 when raised to power 2 will give the same result thereby making give the same solution (image); hence some elements in the domain would have the same image in the co-domain, which is not a property of a One – One mapping.

1. Composite Function: Composite function, which is also known as composition or product of function is the function (say and ), from A into C which assigns each the unique element of i.e. the range of function becomes the domain of function . This relationship is usually written as or as .

Example 3: The functions on the set of real numbers are defined by respectively. Determine the formula for the composite functions: (a) (b) (c)

Solution: (a)

(b)

(c)

Working method: substitute what was equal to in place of starting from right to left, and continue substituting until the last the last is substituted.

Note the following about composite mapping: (i) the operation of the composition of mappings is not commutative i.e. (ii) the operation of the composition of mappings is associative i.e.

CLASS ACTIVITIES:

1. Let the function be defined by the arrow diagrams below :

A B C D

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u

v

x

a

b

c

d

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11

 

1. Determine if each function is Onto.
2. Find the composition writing your result as ordered pairs

1. A relation is defined by g where R is the set of real numbers. Show that g is One – One but not Onto.

PRACTICE EXERCISE:

Objective Test:

1. Two functions are defined by evaluate (a) -49 (b) -47 (c) -10 (d) -9
2. Given that the image of p under the mapping , -10, what is the value of p? (a) -1 (b) -2 (c) -3 (d) -4
3. Find the range of if the domain is .

Essay Test:

1. The mapping defined on the set of real numbers is such that
2. If

the domain and range of f. (ii) is f one to one or onto?

1. Find the range, co-domain and hence determine if it is a one-one or onto function.

A

 

-1

0

1

3

9

0

1

 

1. What is the difference between a mapping and a function?
2. The zeros of a function are the elements of the domain whose images are equal to zero. In other words, determine the zeros of the function defined by:

ASSIGNMENT

1. The zeros of a function are the elements of the domain whose images are equal to zero. In other words, determine the zeros of the function defined by:
2. Determine the domain D of the mapping is the range and f is defined on D.
3. If is a mapping defined on the set R of real numbers. Determine the pre images of (a) 1 (b)-1 (c)7
4. If is a mapping defined on the set of real numbers excluding -2, find g(1), g(0.5), g(-1).
5. Discuss the terms (a) Relation (b) Mapping (c) Functions.

KEY WORDS

• RELATION
• MAPPING
• FUNCTION
• IMAGES
• DOMAIN
• SUB DOMAIN
• ELEMENTS
• RANGE
• ONE-TO-ONE
• ONTO
• COMPOSITE
• CONSTANT
• INVERSE
• ARROW DIAGRAM.