functions and graphs the mathematics of relations

Functions and Graphs

The Mathematics of Relations

Definition of a Relation

Relation

(1)32 mpg(2)8 mpg(3)16 mpg

(A)

(C)

(B)

Domain and Range

• The values that make up the set of independent values are the domain

• The values that make up the set of dependent values are the range.

• State the domain and range from the 4 examples of relations given.

Quick Side Trip Into the Set of Real Numbers

The Set of Real Numbers

Ponder

• To what set does the sum of a rational and irrational number belong?

• How many irrational numbers can you generate for each rational number using this fact?

Properties of Real Numbers• Transitive:

If a = b and b = c then a = c• Identity:

a + 0 = a, a • 1 = a• Commutative:

a + b = b + a, a • b = b • a• Associative:

(a + b) + c = a + (b + c)(a • b) • c = a • (b • c)

• Distributive: a(b + c) = ab + aca(b - c) = ab - ac

Definition of Absolute Value

if a is positive

if a is negative

The Real Number Line

End of Side Trip Into the Set of Real Numbers

Definition of a Relation

• A Relation maps a value from the domain to the range. A Relation is a set of ordered pairs.

• The most common types of relations in algebra map subsets of real numbers to other subsets of real numbers.

Example

Domain Range

3 π

11 - 2

1.618 2.718

Define the Set of Values that Make Up the Domain and Range.

• The relation is the year and the cost of a first class stamp.

• The relation is the weight of an animal and the beats per minute of it’s heart.

• The relation is the time of the day and the intensity of the sun light.

• The relation is a number and it’s square.

Definition of a Function

• If a relation has the additional characteristic that each element of the domain is mapped to one and only one element of the range then we call the relation a Function.

Definition of a Function• If we think of the domain as the set of all

gas pumps and the range the set of cars, then a function is a monogamous relationship from the domain to the range. Each gas pump gets used by one car.

• You cannot put gas in 2 cars as the same time with one pump. (Well not with out current pump design )

x

DOMAIN

y

RANGE

f

FUNCTION CONCEPT

x

DOMAIN

y1

y2

RANGE

R

NOT A FUNCTION

y

RANGE

f

FUNCTION CONCEPT

x1

DOMAIN

x2

Examples

• Decide if the following relations are functions.

X Y

1 2

-5 7

-1 2

3 3

X Y

1 1

-5 1

-1 1

3 1

X Y

1 2

1 7

1 2

1 3

X Y

1 π

π 1 -1 5

π 3

Ponder

• Is 0 an even number?• Is the empty set a function?

Ways to Represent a Function• Symbolic

x,y y 2x or

y 2x

X Y

1 2

5 10

-1 -2

3 6

• Graphical

• Numeric

• VerbalThe cost is twice the original amount.

Example

• Penney’s is having a sale on coats. The coat is marked down 37% from it’s original price at the cash register.

• If you chose a coat that originally costs $85.99, what will the sale price be? What amount will you pay in total for the coat (Assume you bought it in California.)

• Is this a function? What is the domain and range? Give the symbolic form of the function. If you chose a coat that costs $C, what will be the amount $A that you pay for it?

Function NotationThe Symbolic Form

• A truly excellent notation. It is concise and useful.

y f x

y f x • Output Value• Member of the Range• Dependent Variable

These are all equivalent names for the y.

• Input Value• Member of the Domain• Independent Variable

These are all equivalent names for the x.

Name of the function

Example of Function Notation

• The f notation

f x x 1

f 2 2 1

Graphical Representation• Graphical representation of functions

have the advantage of conveying lots of information in a compact form. There are many types and styles of graphs but in algebra we concentrate on graphs in the rectangular (Cartesian) coordinate system.

Average National Price of Gasoline

Graphs and Functions

Domain

Range

CBR

Vertical Line Test for Functions

• If a vertical line intersects a graph once and only once for each element of the domain, then the graph is a function.

Determine the Domain and Range for Each Function

From Their Graph

Big Deal!

• A point is in the set of ordered pairs that make up the function if and only if the point is on the graph of the function.

Numeric

• Tables of points are the most common way of representing a function numerically

Verbal

• Describing the relation in words. We did this with the opening examples.

Key Points

• Definition of a function• Ways to represent a function

SymbolicallyGraphicallyNumericallyVerbally