§ 2.1 Introduction to Functions and sets from 1.1.
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Transcript of § 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
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.1
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
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.1
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}
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 2.1
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.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 2.1
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.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 2.1
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.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 2.1
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.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 2.1
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
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 2.1
Basic Functions 101
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Find the indicated function value:
52 63for 4 zzzQQ
52 46434 Q
102461634 Q
6144484 Q
61924 Q
Replace z with -4
Evaluate the exponents
Multiply
Add
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 2.1
Basic Functions 101
Check Point 3a.Check Point 3a.
SOLUTIONSOLUTION
Find the indicated function value:
54for 6 xxff
5646 f
524
29
Replace x with 6
Multiply
Add and Subtract
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 2.1
Basic Functions 100
Check Point 3b.Check Point 3b.
SOLUTIONSOLUTION
Find the indicated function value:
103gfor 5 2 xxg
10535 2 g
10253
1075
65
Replace x with -5
Evaluate the exponent
Multiply
Add and Subtract
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 2.1
Basic Functions 100
Check Point 3c.Check Point 3c.
SOLUTIONSOLUTION
Find the indicated function value:
27for 4 2 rrrhh
24744 2 h
2)4(716
22816
46
Replace r with -4
Evaluate the exponent
Multiply
Add and Subtract
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 2.1
Basic Functions 101
Check Point 3d.Check Point 3d.
SOLUTIONSOLUTION
Find the indicated function value:
96Ffor xxhaF
96 hahaF
966 ha
Replace x with a+h
Multiply
NOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!
DONE
Blitzer, Intermediate Algebra, 5e – Slide #21 Section 2.1
Basic Functions 101
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Find the indicated function value:
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!!!