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CHARACTERISTICS OF RELATIONS AND FUNCTIONS (Unit1.1a)

Introduction: A relation is a correspondence between two sets of numbers for which

each element in the initial set (the source) is mapped to one or sometimes more than one

element in the final set (the destination).

The set of elements that represent the source is called the Domain and is identified as (X)

and the set that represents the destination is called the Range and is identified as (Y).

This correspondence is typically represented by an arrowed link whereby only the

terminal (final) end of the link (at the destination set) is fitted with the arrow-head.

Source(X) Destination(Y)

If the each element in the Domain (X) corresponds to only one element in the Range

(Y) then this mathematical relation is said to be a functional relation or just simply called

a function.

Examples of Mapping Diagrams:-

A  B 

Exercise: Which of the above diagrams represents a function? Answer ( A / B )

The correspondence/mapping from the domain set (X) to the range set (Y) could be

displayed using a number of diagrams as shown below

Exercise: Could you name the following type of diagrams and state if it is a function.

Type of diagram: ________ function (Y/N) ; Type of diagram: _______ function (Y/N)

Ordered pairing: ( x , y )

x

y

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Learning Outcomes

  introduced to the relevant terminology

  determine the domain and range of a relations 

  determine whether a relation is a function

 

work with different representations of a function namely mapping diagrams,tables, ordered pairs, equations and graphs 

Understanding the Terminology

TERM DEFINITION EXAMPLES

Relation It represents a correspondence between

two sets of values namely set X and set Y 

x -2 -1 0 0.9 1 1.1 2

y

A. Mapping Diagram

B. Mapping Diagram

X Y

0

1

4

9

-3

-2

-1

0

1

2

3

C. Table of values

X 1 2 3 4

Y 1 1/2 1/3 1/4

D. Co-ordinates

Set of ordered pairs (x, y) :-

{ (1,1), (2,3), (4,5), (2,8), (3,1)}

E. Algebraic Expression

=1

− 1 

X Y

0

123

0

246

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TERM DEFINITION EXAMPLES

Domain (D) Set of all permissible values of X in any

relation

Based upon the relations on pg.2 :-

 A.  D  = { 0 ,1, 2, 3}

Complete the following:- B.

  D  = { }

C.  D  = { }

D.  D  = { } 

E.  D  = { } 

Range (R) Set of all permissible values of Y

(obtained from permissible values of X)

Based upon the relations on pg.2 :-

 A.  R  = { 0 ,2, 4, 6}

Complete the following:-

 B. 

R  = { }C.  R  = { }

 D. R   = { }

 E. R   = { }

Function ( f(x) ) It represents a correspondence between

two sets of elements namely set X and set

Y such that each element in the first set

(X) corresponds to one and only one

element in the second set (Y)

Question : Using the terminology shown at

the top of the page, identify the name for

the first set ( X  ) : ___________

and second set ( Y  ) : ___________

Corollary of the definition

If a function is represented via a set of

ordered pairs (i.e.: (x,y)) then no two pairs

with identical x values are allowed tohave different y values.

i.e.: { ... (2,4),... (2,5),....} ≠ function 

Based upon the relations on pg.2 :-

A.  is a function but

B.  is not a function

Why?

 __________________________

 __________________________

Using the criteria in the corollary,

determine whether C. , D. and E.

would represent functions.

C.

 D.

 E.

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Common and interchangeable representations of functions

Depending on the required analysis, functions could be represented using three common

approaches namely, the Graphical approach, Numerical approach and the Algebraic

approach.

Example:

Numerical Graphical Algebraic

X Y

-2 3

-1 0

0 -1

1 0

2 3

y

4

3

2

1

-2 -1 0 1 2 x 

-1

y = x2  –  1

If the emphasis is on displaying multiple properties of the function then a graphical approach

would be advisable. However, a numerical approach is adopted when it is important to show

the precise correspondence between the values in the domain (X) and the values in the range

(Y). Typical numerical approaches are the mapping diagram, ordered pairs or tables.

An algebraic approach is used when summarizing the relationship between all domain and

range values. This technique is most suitable when adapting functions to suit application-type

 problems. (to be discussed in later topics)

 Exercise:

 Demonstrate that each of the above illustrations in the example are in- fact equivalent

representations of one another.

[Hint : Beginning with the Numerical representation, convert all key values in ordered pairs

 for each of the respective representations. Compare and then conclude.]

Comparing the Ordered Pairing of key values  for each of the above representation

 Numerical:

{ (-2, 3), }

Graphical:

{ }

 Algebraic:

{ }

Conclusion:

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HOMEWORK (Text pg. 11 to 13)

Question 1

Question 3

Question 5

Question 8

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Question 6

Question 7