Relations and FunctionsLesson 1: Properties of Functions

Todays Objectives Explain, using examples, why some

relations are not functions but all functions are relations

Determine if a set of ordered pairs represents a function

Generalize and explain rules for determining whether graphs and sets of ordered pairs represent functions

Relations and Functions A relation associates the elements of one

set of data with the elements of another set of data

A function is a specific type of relation where each element in the domain is associated with exactly one element in the range. On a graph, this means that each x-value has exactly one corresponding y-value. Look at the graph on the next page for example:

Relation vs. Function

Vertical Line Test One simple method used to determine if a

graph is a relation or a function is the vertical line test.

If you draw a vertical line at any point on a coordinate plane, and it intersects the graph only once, it is a function

If it intersects the graph more than once, it is not a function

All functions are relations, but not all relations are functions

Is this a function?

Is this a function?

Is this a function?

Is this a function?

Is this a function?

Is this a function?

Domain and Range The set of first elements of a relation is

called the domain. When graphing a relation, the set of first elements will be the x-values of the graph.

The set of related second elements of a relation is called the range. When graphing a relation, the set of second elements will be the y-values of the graph.

Example: Arrow Diagram Here are two relations that relate vehicles to

the number of wheels each has:

1234

BicycleCarMotorcycleTricycleUnicycle

BicycleCarMotorcycleTricycleunicycle

1234

is the number of wheels on a has this number of wheels------------------------------- ----------------------------

x y x y

This diagram does NOT represent a function because there is one element in the first set that is related with two elements in the second set

This diagram DOES represent a function because each element in the first set associates with exactly one element in the second set

Example: Ordered Pairs 

{(1, unicycle), (2, bicycle), (2,

motorcycle), (3, tricycle), (4,

car)}

 

The set of ordered pairs above

does not represent a function

because two ordered pairs have

the same first element.

The domain is the set of first

elements: {1,2,3,4}

 

The range is the set of

associated second elements:

{unicycle, bicycle, motorcycle,

tricycle, car}

 

{(unicycle, 1), (bicycle, 2),

(motorcycle, 2), (tricycle, 3),

(car, 4)}

The set of ordered pairs above

represents a function because

the ordered pairs have

different first elements

The domain is the set of first

elements: {unicycle, bicycle,

motorcycle, tricycle, car}

The range is the set of

associated second elements:

{1,2,3,4}

Example: You doWith a partner, try the Check

your understanding question on page 266 of the textbook

Example: Table of Values

Hours worked, h Gross Pay, P($)

1 12

2 24

3 36

4 48

5 60

In many jobs, a person’s pay, P dollars, often depends on the number of hours worked, h. Therefore, we say that P is the dependent variable. The number of hours worked, h, does NOT depend on the pay, P, so we say that h is the independent variable.

The values of the independent variable are listed first, and these elements belong to the domain.The values of the dependent variable are listed second, and these elements belong to the range.

Dom

ain Ran

ge

Example: You doWith a partner, try the Check

your understanding question on page 267 of the textbook

Function Notation Function notation is a way of showing the

independent variable in a function; for example, f(x) means that the value of the function f depends on the value of the independent variable x.

In the last example, the value of the gross pay, P, depends on the value of the independent variable (hours worked), h, so in function notation it would be written as P(h).

We say these as “f of x” and “P of h”

Example: You doWith a partner, try the Check

your understanding question on page 269-270 of the textbook

HomeworkPage 270-273, #4, 6, 9, 10, 12, 14-16, 18-19, 22.