Chapter 2 Functions and Linear Functions. § 2.1 Introduction to Functions.
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Transcript of Chapter 2 Functions and Linear Functions. § 2.1 Introduction to Functions.
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 =