§ 2.1 introduction to functions and sets from 1.1

§ 2.1

Introduction to Functions and sets from 1.1

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 1.1

Sets p 6

The braces, { }, indicate that we are representing a set. This form of representation, called the roster method, uses commas to separate the elements of the set. The ellipsis indicates that there is no final element and that

the listing goes on forever.

The objects in a set are called the elements The objects in a set are called the elements of the set. Such as:of the set. Such as:

A set is a collection of objects whose contents A set is a collection of objects whose contents can be clearly determined.can be clearly determined.

,5,4,3,2,1


Set-Builder Notation p 6

{x | x is a real number and greater than 10}

Express x > 10 using set-builder notation

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

The set of all x Such that X is a real number greater than 10


Symbols p 7 and

The symbol is used to indicate that a number or object is ina particular set. Here is an example:

7 {1,2,5,7,9}

The symbol is used to indicate that a number or object is not in a particular set. For example:

3 {4,6}


Relation p 96

Definition of a Relation

A relation is any set of ordered pairs.

The set of all first components of the ordered pairs is called the domain of the relation

and the set of all second components is called the range of the relation.


Relation p 96

Check Point 1Find the domain and the range of the relation: {(0, 9.1), (10, 6.7), (20, 10.7), (30, 13.2), (34, 15.5)}

Domain: {0, 10, 20, 30, 34}

Range: {9.1, 6.7, 10.7, 13.2, 15.5}

What is the rule that assigned the “inputs” in the domain to the “outputs” in the range? For example, for the ordered pair (30, 13.2), how does the data in Figure 2.1(a), which shows the percentage of first-year US college students claiming no religious affiliation.

010

3020

6.710.7

9.1

13.2

Domain Range34 15.5


Functions p 98

Definition of a Function

A function is a correspondence from a first set, called the domain, to a second set, called the range, such that each element in the domain corresponds to exactly one element

in the range.

(Each x corresponds to exactly one y in a function. Values of x have to be “faithful” – so to speak – that is, each x goes to exactly one y. However, a y value

may be the image of more than one x – so y’s are not required to be “faithful”)

For example, the following relation is a function:

{(1,2), (2,3), (3,5), (4,3)} Note that the y value 3 is the image of two x values.


Basic Functions 98

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Determine whether the following is a function.

12

43

guitarviolin

drums

flute

Yes, because none of the members of the domain correspond to more than one member of the range.

Domain Range


Basic Functions 98

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Determine whether the following is a function.

38

95

beetlescrickets

ants

moths

No, because one of the members of the domain, 9, corresponds to more than one member of the range.

Domain Range


Functions 99

Example: Determine whether each relation is a function:

(a) {(2,3), (4,5), (6,5), (9,10)}

(a)Yes – a function

(b) {(1,2), (3,3), (6,8), (1,10)}

(b) No – not a function since 1 is mapped to 2 and 10

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

(c) Yes – a function


Relation p 99

Check Point 2Determine whether each relation is a function:

a. {(1,2), (3, 4), (5, 6), (5, 7)}

b. {(1, 2), (3, 4), (6, 5), (7, 5)}

Not a function

function


Functions as Equations 99

Functions are usually given in terms of equations rather than as sets of ordered pairs.

For example, consider the function y = 2x – 3

For each value of x, there is one and only one value of y.

The variable x is called the independent variable and the variable y is called the dependent variable.


Function Notation 100

If an equation in x and y gives only one value of y for each value of x, then the variable y is a function of the variable x.

When an equation represents a function, the function is often named by a letter such as f, g, h, F, G, or H.

The output value, the y value, is often denoted by f(x), read

“f of x” or “f at x”

The notation f(x) does not mean “f times x”. The notation describes the value of the function f at x.


Basic Functions 100

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the indicated function value:

724for 3 3 xxxff

732343 3 f

7322743 f

761083 f

1073 f

Replace x with 3

Evaluate the exponent

Multiply

Add and Subtract


Basic Functions 101

EXAMPLEEXAMPLE

SOLUTIONSOLUTION


52 63for 4 zzzQQ

52 46434 Q

102461634 Q

6144484 Q

61924 Q

Replace z with -4

Evaluate the exponents

Multiply

Add


Basic Functions 101

Check Point 3a.Check Point 3a.

SOLUTIONSOLUTION


54for 6 xxff

5646 f

524

29

Replace x with 6

Multiply

Add and Subtract


Basic Functions 100

Check Point 3b.Check Point 3b.

SOLUTIONSOLUTION


103gfor 5 2 xxg

10535 2 g

10253

1075

65

Replace x with -5


Multiply

Add and Subtract


Basic Functions 100

Check Point 3c.Check Point 3c.

SOLUTIONSOLUTION


27for 4 2 rrrhh

24744 2 h

2)4(716

22816

46

Replace r with -4


Multiply

Add and Subtract


Basic Functions 101

Check Point 3d.Check Point 3d.

SOLUTIONSOLUTION


96Ffor xxhaF

96 hahaF

966 ha

Replace x with a+h

Multiply

NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!


Basic Functions 101

EXAMPLEEXAMPLE

SOLUTIONSOLUTION


wwwLyxL 63for 2

yxyxyxL 63 2

yxyxyxyxL 63

yxyxyxyxyxL 63 22

Replace w with x+y

Rewrite exponent

Multiply

Add yxyxyxyxL 623 22

yxyxyxyxL 6663 22 Distribute

NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!

§ 2.1 introduction to functions and sets from 1.1

Documents

e slide

intermediate algebra

relation p

set of ordered pairs

setbuilder notation

ina particular set

functions p

following relation