Inequality Symbol Number Line Graph Interval Notation

MA 15910, Lessons 2a and 2b Introduction to Functions Algebra: Sections 3.5 and 7.4 Calculus: Sections 1.2 and 2.1

Representing a Set of Numbers

Inequality Symbol Number Line Graph Interval Notation

xa

)

a

xa ( a

(, a) (a, )

xa

]

(, a]

a

x a

[

a

[a, )

a xb

(

a

)

(a, b)

b

axb [ a

axb ( a

a x b [a

no solution

)

[a, b)

b

]

(a, b]

b

]

[a, b]

b

all real numbers

(, )

You are responsible for knowing all 3 ways above to represent a set of numbers that is an inequality; with the inequality symbol, graphed on a number line, or in interval notation.

1

Example 0: (a) Represent the set of numbers < 12, using a number line and interval notation.

(b) Represent the set of numbers 2, using a number line and interval notation.

(c) Represent the set of numbers -5 < 0, using a number line and interval notation.

(d) Represent the set of numbers (3, ), using an inequality symbol and using a number line.

(e) Represent the set of numbers (-, -100], using an inequality symbol and using a number line.

(f) Represent the numbers represented on the number line below in both interval notation and using an inequality symbol.

[

)

-1

7

Definition: A relation is any set of ordered pairs. The set of first components in the ordered pairs is called the domain of the relation. The set of second components is called the range of the relation. Relations may be represented as sets, tables, diagrams, graphs, or equations.

Definition: A function is a relation in which no two ordered pairs have the same first components but different second components. Each element or component of the domain (input values) is paired to one and only one element or component of the range (output values).

2

Relations or functions can be represented by sets of ordered pairs, tables of ordered pairs, mappings of ordered pairs, graphs, or equations (as seen in the next few examples).

Example 1: Which relations below represent functions? State the domains and ranges.

a) {(9,81), (4,16), (5,25), (2,4), (6,36)} Function?

Domain:

Range:

b) x y

-3 4 2 -1

Domain:

12 9 0 2

Function? Range:

c)

5 3 -2

DOMAIN

12 -1

RANGE

Function?

d)

2 2

(-3,-3)

e) x y2

(1,-1) (3,-1)

Function? Domain:

Function? Domain: Range:

Range:

(continued on the next page) 3

f ) y 2x 5 x3

Function?

g) s 3r 2 Function?

Domain: Domain:

Example 2: Which below represent functions? What is the domain? What is the range?

h) {(2,6), (7,8), (2,0)}

i) x q(x) 2 5 3 5 9 5 0 5

j)

3

DOMAIN

4

k)

8

RANGE

4

Function Notation: Functions can be `named' by using letters. This `name' can be used to write the function. For example; the function h represented by y 2x x2

can be written as h(x) 2x x2 . This type of notation is known as function notation. The element

of the domain, the input (the x) is inside the parentheses and the result, the output, (the y or h(x)) is the matching element of the range.

Example 2: Given the function g(x) 3x 1 , find the following function values.

a) g(2)

b) g(0)

c) g(4)

d) g( 2)

Example 3: Given the functions f (x) x2 2x and F(x) x , find the following. 3x 1

a) f (3x)

g) F (3x)

b) F (a 1)

h) f (a 1)

c) f (r 2)

i) F (r 2)

d)

F

1 b

e) f (x) f (2)

j)

f

1 b

k) F (x) F (2)

f ) f (x h) f (x)

5

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