1.5 I understand Functions

Function

• For each input there is only one output• Ex: Let g represent the age of a US citizen and

d represent the number of days since his or her birth. Is g a function of d or is d a function of g?

If y is a function of x

• Independent variable: x • Dependent variable: y

• The DETERMINES the value of .

Relation

• A set of ordered pairs• Can be indicated by a verbal description, a

graph, a formula or equation, or a table• Not every relation is a function, but every

function is a relation.

Determine if each relation is a function:

•

X Y

1 9

20 29

6 10

-1 9

X Y

5 7

6 8

7 9

5 6

Vertical Line Test

• A graph is a function if and only if there is no vertical line that crosses the graph more than once

• AKA: When you draw a vertical line on the graph, if it intersects with the graph more than once, it is NOT a function

Use the Vertical Line Test to determine if the graph is a function

Identifying a function from an equation

•

Domain and Range

•

Determine the Domain and Range (Put into calculators if needed)

•

12 xy

Function Notation

• A function defined by a set of ordered pairs can be named with a letter

= { (2,5), (3,8)}Since pairs 2 with 5, we write f(2)=5The value at 2 is 5.f(3)=8; The value at 3 is 8.

Using Function Notation

• 3)( xxf

Using Function Notation with Variables

• 2)( 2 xxf

The Average Rate of Change

• Slope =

• It is not necessary to have a formula for a function to find the average rate of change, you can find the average rate of change (slope) if you know two points on the graph.

12

12

xx

yy

Average Rate of Change (slope)

• Find the average rate of change between the two points.

Example 1

• The population of California was 29.8 million in 1990 and 33.9 million in 2000. What was the average rate of change of the population over that time interval?

• Use the coordiantes (year, population) (1990, 29,800,000) and (2000, 33,900,000)

The Difference Quotient

h

xfhxf )()(

Find and simplify the difference quotient

•

Find and simplify the difference quotient

•

Find and simplify the difference quotient

xxf )(

1.6 Graphs of Relations and Functions

• Parent Functions• Graph the following parent functions and

state the domain and range.

X Y

X Y

X Y

The Absolute Value Function

The square root function:

xy

xy

X Y

The Cubic function:

X Y

3xy

X Y

X Y

X Y

X Y