Chapter 2

Functions and Linear Functions

§ 2.1

Introduction to Functions

Blitzer, Intermediate Algebra, 4e – Slide #3

Basic Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Determine whether the following is a function.

12

43

guitarviolin

drums

flute

Yes, because none of the members of the domain correspond to more than one member of the range.

Domain Range

Blitzer, Intermediate Algebra, 4e – Slide #4

Basic Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Determine whether the following is a function.

38

95

beetlescrickets

ants

moths

No, because one of the members of the domain, 9, corresponds to more than one member of the range.

Domain Range

Blitzer, Intermediate Algebra, 4e – Slide #5

Basic Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the indicated function value:

724for 3 3 xxxff

732343 3 f

7322743 f

761083 f

1073 f

Replace x with 3

Evaluate the exponent

Multiply

Add and Subtract

Blitzer, Intermediate Algebra, 4e – Slide #6

Basic Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the indicated function value:

52 63for 4 zzzQQ

52 46434 Q

102461634 Q

6144484 Q

61924 Q

Replace z with -4

Evaluate the exponents

Multiply

Add

Blitzer, Intermediate Algebra, 4e – Slide #7

Basic Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the indicated function value:

wwwLyxL 63for 2

yxyxyxL 63 2

yxyxyxyxL 63

yxyxyxyxyxL 63 22

Replace w with x+y

Rewrite exponent

Multiply

Add yxyxyxyxL 623 22

yxyxyxyxL 6663 22 DistributeNOTE: THIS CANNOT BE SIMPLIFIED ANY FURTHER!!!

Blitzer, Intermediate Algebra, 4e – Slide #8

Graphing Functions

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Graph the function . 13 xxf

x y = -3x+1 Ordered Pair (x,y)-2 f(x) = -3(-2) + 1 = 6 + 1 = 7 (-2,7)

-1 f(x) = -3(-1) + 1 = 3 + 1 = 4 (-1,4)

0 f(x) = -3(0) + 1 = 0 + 1 = 1 (0,1)

1 f(x) = -3(1) + 1 = -3 + 1 = -2 (1,-2)

2 f(x) = -3(2) + 1 = -6 + 1 = -5 (2,-5)

Blitzer, Intermediate Algebra, 4e – Slide #9

Graphing Functions

CONTINUECONTINUEDD

Blitzer, Intermediate Algebra, 4e – Slide #10

The Vertical Line Test

EXAMPLEEXAMPLE

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

(a) (b) (c)

Blitzer, Intermediate Algebra, 4e – Slide #11

The Vertical Line Test

SOLUTIONSOLUTION

CONTINUECONTINUEDD

(a) (b) (c)

y is a function of x y is not a function of x y is a function of x

Blitzer, Intermediate Algebra, 4e – Slide #12

Graphs of Functions

EXAMPLEEXAMPLE

The figure shows the cost of mailing a first-class letter, f(x), as a function of its weight, x, in ounces. Use the graph to answer the following questions.

Blitzer, Intermediate Algebra, 4e – Slide #13

Graphs of Functions

(a) Find f (3). What does this mean in terms of the variables in this situation?

CONTINUECONTINUEDD

(b) Find f (4). What does this mean in terms of the variables in this situation?

(c) What is the cost of mailing a letter that weighs 1.5 ounces?

(d) What is the cost of mailing a letter that weighs 1.8 ounces?

Blitzer, Intermediate Algebra, 4e – Slide #14

Graphs of Functions

SOLUTIONSOLUTION

CONTINUECONTINUEDD

f (3) = 0.83. This means that when a first-class letter weighs 3 ounces, postage costs 83 cents.

(a) Find f (3). What does this mean in terms of the variables in this situation?

Blitzer, Intermediate Algebra, 4e – Slide #15

Graphs of Functions

CONTINUECONTINUEDD

f (4) = 1.06. This means that when a first-class letter weighs 4 ounces, postage costs $1.06.

(b) Find f (4). What does this mean in terms of the variables in this situation?

Blitzer, Intermediate Algebra, 4e – Slide #16

Graphs of Functions

CONTINUECONTINUEDD

f (1.5) = 0.60. This means that when a first-class letter weighs 1.5 ounces, postage costs $0.60.

(c) What is the cost of mailing a letter that weighs 1.5 ounces?

Blitzer, Intermediate Algebra, 4e – Slide #17

Graphs of Functions

CONTINUECONTINUEDD

f (1.8) = 0.60. This means that when a first-class letter weighs 1.8 ounces, postage costs $0.60.

(d) What is the cost of mailing a letter that weighs 1.8 ounces?

Blitzer, Intermediate Algebra, 4e – Slide #18

Graphs of Functions

Obtaining Information from GraphsA 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.

Blitzer, Intermediate Algebra, 4e – Slide #19

Domain and Range

EXAMPLEEXAMPLE

Use the graph of the function to identify its domain and range.

Blitzer, Intermediate Algebra, 4e – Slide #20

Domain and Range

{x | x is a real number}

SOLUTIONSOLUTION

{y | y 3.6}

Domain =

Range =