Section 2.1Functions and Graphs

Relations

Domain: sitting, walking, aerobics, tennis, running, swimming

Range: 80,325,505,720,790

Do not list 505 twice.

Example

Find the domain and the range.

98.6, Felicia , 98.3,Gabriella , 99.1, Lakeshia

Functions

A relation in which each member of the domain corresponds to exactly one member of the range is a function. Notice that more than one element in the domain can correspond to the same element in the range. Aerobics and tennis both burn 505 calories per hour.

Is this a function? Does each member of the domain correspond to precisely one member of the range? This relation is not a function because there is a member of the domain that corresponds to two members of the range. 505 corresponds to aerobics and tennis.

Example

Determine whether each relation is a function?

1,8 , 2,9 , 3,10

2,3 , 2,4 , 2,5

3,6 , 4,6 , 5,6

Functions as Equations

2

Here is an equation that models paid vacation days each

year as a function of years working for the company.

y=-0.016x .93 8.5

The variable x represents years working for a company.

The variable y re

x

presents the average number of vacation days

each year. The variable y is a function of the variable x.

For each value of x, there is one and only one value of y.

The variable x is called the independent variable because

it can be assigned any value from the domain. Thus, x can

be assigned any positive integer representing the number

of years working for a company. The variable y is called the

dependent variable because its value depends on x. Paid

vacation days depend on years working for a company.

Not every set of ordered pairs defines a function.

Not all equations with the variables x and y define

a function. If an equation is solved for y and more

than one value of y can be obtained for a given x,

then the equation does not define y as a function of x.

So the equation is not a function.

Example

Determine whether each equation defines y as a function of x.

2

2 2

4 8

2 10

16

x y

x y

x y

Function Notation

The special notation f(x), read "f of x" or "f at x"

represents the value of the function at the number x.

If a function named f, and x represents the independent

variable, the notation f(x) corresponds

2

2

to the y-value for

a given x.

f(x)=-0.016x .93 8.5

This is read "f of x equals -0.016x .93 8.5"

x

x

2

We are evaluating the function at 10 when we

substitute 10 for x as we see below.

(10) -0.016 10 .93 10 8.5

What is the answer?

f

Graphing Calculator- evaluating a function

1

Press the VARS key. Move the cursor to

the right to Y-VARS. Press ENTER on 1.

Function. Press ENTER on Y .

Type (10) then ENTER. You will now see the

same answer that you saw on the previous

screen when you evaluated the equation at x=10.

l

2

Press the Y = key. Type in the equation

f(x)= - 0.016x .93 8.5

Quit this screen by pressing 2nd Mode (Quit).

x 2( ) .016 x,T, ,n x .93 x,T, ,n + .85

Example

Evaluate each of the following.

2

2

Find f(3) for f(x)=2x 4

Find f(-2) for f(x)=9-x

Example

Evaluate each of the following.

2

2

Find f(x+2) for f(x)=x 2 4 ?

Is this is same as f(x) + f(2) for f(x)=x 2 4

x

x

Example

Evaluate each of the following.

2

2

Find f(-x) for f(x)=x 2 4

Is this is same as -f(x) for f(x)=x 2 4?

x

x

Graphs of Functions

The graph of a function is the graph of its ordered pairs.

First find the ordered pairs, then graph the functions.

Graph the functions f(x)=-2x; g(x)=-2x+3

x f(x)=-2x (x,y) g(x)=-2x+3 (x,y)

-2 f(-2)=4 (-2,4) g(-2)=7 (-2,7)

-1 f(-1)=2 (-1,2) g(-1)=5 (-1,5)

0 f(0)=0 (0,0) g(0)=3 (0,3)

1 f(1)=-2 (1,-2) g(1)=1 (1,1)

2 f(2)=-4 (2,-4) g(2)=-1 (2,-1)

See the next slide.

x

y

x

y

f(x)g(x)

Example

Graph the following functions f(x)=3x-1 and g(x)=3x

x

y

The Vertical Line Test

x

y

The first graph is a function, the second is not.

x

y

x

y

Example

Use the vertical line test to identify graphs in which y is a function of x.

x

y

Example

Use the vertical line test to identify graphs in which y is a function of x.

x

y

x

y

Obtaining Information

from Graphs

You can obtain information about a function from its graph.

At the right or left of a graph you will find closed dots,

open dots or arrows.

A closed dot indicates that the graph does not extend

beyond this point, and the point belongs to the graph.

An open dot indicates that the graph does not

extend beyond this point and the point

does not belong to the graph.

An arrow indicates that the graph extends

indefinitely in the direction in

which the arrow points.

Example

Analyze the graph.2( ) 3 4

a. Is this a function?

b. Find f(4)

c. Find f(1)

d. For what value of x is f(x)=-4

f x x x

x

y

Identifying Domain and Range from a Function’s Graph

x

yIdentify the function's domain and range from the graph

Domain (-1,4]

Range [1,3)

Domain [3, )

Range [0, )

x

y

Example

Identify the Domain and Range from the graph.

x

y

Example

Identify the Domain and Range from the graph.

x

y

Example

Identify the Domain and Range from the graph.

x

y

Identifying Intercepts

from a Function’s Graph

We can identify x and y intercepts from a function's graph.

To find the x-intercepts, look for the points at which the graph

crosses the x axis. The y-intercepts are the points where the graph

crosses the y axis.

The zeros of a function, f, are the x values for which f(x)=0.

These are the x intercepts.

By definition of a function, for each value of x we can

have at most one value for y. What does this mean in terms

of intercepts? A function can have more than one x-intercept

but at most one y intercept.

Example

Find the x intercept(s). Find f(-4)

x

y

Example

Find the y intercept. Find f(2)

x

y

x

y

Example

Find the x and y intercepts. Find f(5).

(a)

(b)

(c)

(d)

Find f(7).

x

y

0

1

1

2

(a)

(b)

(c)

(d)

Find the Domain and Range.

D:(- , ) R:(-5,7]

D:(-5, ) R: (- , )

D:(- , ) R: [-5, )

D:[- , ] R: [-5, ]

x

y

(a)

(b)

(c)

(d)

22 3( ) Find f(-1)

7

xf x

1

7

57

1

71