Date post: | 21-Dec-2015 |
Category: |
Documents |
Upload: | lee-chapman |
View: | 229 times |
Download: | 3 times |
Chapter 2
Functions and Graphs
Section 1
Functions
2Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 2.1Functions
The student will be able to do point-by-point plotting of equations in two variables.
The student will be able to give and apply the definition of a function.
The student will be able to identify domain and range of a function.
The student will be able to use function notation. The student will be able to solve applications.
3
Graphing Equations
If you are not familiar with a graphβs βfamilyβ, then use point-by-point plotting. (i.e. make an x-y table)β’ However, this is a very tedious process.
Knowing a graphβs family, will help you determine its basic shape.
Knowing a graphβs basic shape and the transformations on its parent, will help you graph it without making an x-y table.β’ This will be reviewed in tomorrowβs lesson.
Barnett/Ziegler/Byleen Business Calculus 12e
4
Families and Shapes
Line
Parabola βV-shapedβ
Barnett/Ziegler/Byleen Business Calculus 12e
5
Families and Shapes
Barnett/Ziegler/Byleen Business Calculus 12e
6Barnett/Ziegler/Byleen Business Calculus 12e
Functions
A relation (set of ordered pairs) represents a function if for each x, there is only one y.
The set of all xβs is called the domain, and the set of all corresponding yβs is called the range.
Which of these relations is a function?β’ {(1, 3), (4, 9), (7, 15), (10, 21)}β’ {(2, 4), (-2, 4), (3, 9), (-3, 0)}β’ {(16, 4), (16, -4), (9, 3), (9, -3)}β’ Answer: The first two are functions.
7Barnett/Ziegler/Byleen Business Calculus 12e
Vertical Line Test for a Function
If you have the graph of an equation, you can easily determine if it is the graph of a function by doing the vertical line test.
8Barnett/Ziegler/Byleen Business Calculus 12e
Vertical Line Test for a Function(continued)
This graph fails the vertical line test, so itβs not a function.
This graph passes the vertical line test, so it is a function.
9Barnett/Ziegler/Byleen Business Calculus 12e
Function Notation
The following notation is used to describe functions. The variable y will now be called f (x).
This is read as β f of xβ and simply means the y coordinate of the function corresponding to a given x value.
can now be expressed as
2 2y x
2( ) 2f x x
10Barnett/Ziegler/Byleen Business Calculus 12e
Function Evaluation
Consider our function Evaluate:
β’ f (β3)
(-3)2 β 2 = 7β’ f(a)
a2 - 2β’ f(2x)
(2x)2 β 2 = 4x2 β 2β’ f(x + h)
(x + h)2 β 2 = x2 + 2xh + h2 β 2
2( ) 2f x x
11Barnett/Ziegler/Byleen Business Calculus 12e
More Examples
f (6 h) 3(6 h) 2 18 3h 2
16 3h
f (2) 3(2) 2 4 2
f (x) 3x 2
12Barnett/Ziegler/Byleen Business Calculus 12e
Domain of a Function
The domain of a function refers to all the possible values of x that produce a valid y.
The domain can be determined from the equation of the function or from its graph.
13Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Domain of a Function
ΒΏπ·πππππ :
π¦=(π₯β 4)3 π¦=β(π₯+1)2 β3 π¦=|π₯β2|+5
If a function does not contain a square root or a denominator then its domain is all reals (-, )
14
Determining Domain
If a function contains a square root or a denominator containing x, its domain will be restricted.
The next few examples show how to determine the restricted domain.
Barnett/Ziegler/Byleen Business Calculus 12e
15Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Domain of a Function
( ) 3 2f x x 3 π₯β 2β₯ 0
π₯β₯23
Functions with square roots:Set the expression inside the
square root 0 and solve for x to determine the domain.
[ 23
, β )π·πππππ :
16Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Domain of a Function
Example: Find the domain of the function
1( ) 4
2f x x
12π₯β 4 β₯ 0
π₯β₯ 8
[ 8 , β )π·πππππ :
17Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Domain of a Function
Functions with x in the denominator:β’ Set the denominator 0 and solve for x
to determine what x cannot be equal to.
1( )
3 5f x
x
3 π₯β5β 0
π₯β 53
(β β ,53 )βͺ( 5
3, β )π·πππππ :
18Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Domain of a Function
π₯β 3
π (π₯ )= 4
β2 π₯+5
Find each domain:
π (π₯ )= 72 (π₯β 3 ) (π₯+2 ) 2 π₯+5>0
π₯>β52
(β β , β2 )βͺ (β2,3 )βͺ (3 ,β )
(β 52
, β )π·πππππ :
π·πππππ :
π₯β3 β 0 π₯+2β 0π₯β β2
β2 3
π (π₯ )= 7
2 π₯2β2 π₯β12
19
Business Analysis
Types of relations involving business applications:β’ Total Costs = fixed costs + variable costs
C = a + bx (linear relation)β’ Price-Demand function = the price for which an item
should be sold when you know the demand
p = m β nx (linear relation)β’ Price-Supply function (similar to above)β’ Revenue = number of items sold price per item
R = xp = x(m β nx) (quadratic relation)β’ Profit = Revenue β Cost
P = x(m β nx) β (a + bx) (quadratic relation)
Barnett/Ziegler/Byleen Business Calculus 12e
20Barnett/Ziegler/Byleen Business Calculus 12e
Mathematical Modeling
The price-demand function for a company is given by
where x represents the number of items and p(x) represents the price of the item.
A) Determine the revenue function.
B) Find the revenue generated if 50 items are sold.
C) What is the domain of the revenue function?
( ) 1000 5 , 0 100p x x x
21Barnett/Ziegler/Byleen Business Calculus 12e
Solution
A) Revenue = Quantity Price R(x) = x β p = x(1000 β 5x)
R(x) = 1000x β 5x2
B) When 50 items are sold, we set x = 50:
C) The domain of the function is the same as the domain for the price-demand function (which was given):
0 100x
π (π₯ )=1000 (50)β 5(50)2
π (50 )=$37,500
ππ [0 ,100 ]
22Barnett/Ziegler/Byleen Business Calculus 12e
Break-Even and Profit-Loss Analysis
Any manufacturing company has costs C and revenues R. They determine the following:
β’ If R < C loss β’ If R = C break even β’ If R > C profit
23Barnett/Ziegler/Byleen Business Calculus 12e
Example of Profit-Loss Analysis
A company manufactures notebook computers. Its marketing research department has determined that the data is modeled by the price-demand functionp(x) = 2,000 β 60x, when 1 < x < 25,
(x is in thousands, p(x) is in dollars).
A) What is the price per computer when the demand is 20 thousand computers?
B) What is the companyβs revenue function and what is its domain?
C) How much revenue is generated for 20 thousand computers?
24Barnett/Ziegler/Byleen Business Calculus 12e
Answer to Revenue Problem
B) Revenue = Quantity Price
The domain of this function is the same as the domain of the price-demand function, which is [1, 25] (in thousands.)
π (π₯ )=π₯ βπ (π₯)π (π₯ )=π₯ (2000 β60 π₯)π (π₯ )=2000 π₯β 60π₯2
A ΒΏπ (20 )=2000 β60 (20)π (20 )=$ 800ππππππ‘πππππ(when the demand is 20,000)
C ΒΏ π (20 )=2000 (20 ) β60 (20)2
π (20 )=16000 (ππ hπ‘ ππ’π ππππ ππ πππππππ )The revenue is $ 16,000,000 for 20,000 notebooks .
25Barnett/Ziegler/Byleen Business Calculus 12e
Profit Problem
The financial department for the company in the preceding problem has established the following cost function for producing and selling x thousand notebook computers:
C(x) = 4,000 + 500x
x is in thousands, C(x) is in thousands of dollars
A) Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function.
B) Does the company make a profit/loss if 20 thousand notebooks are made and sold?
26Barnett/Ziegler/Byleen Business Calculus 12e
Answer to Profit Problem
A) Since Profit = Revenue β Cost, and our revenue function from the preceding problem was R(x) = 2000x β 60x2,
P(x) = R(x) β C(x) = 2000x β 60x2 β (4000 + 500x) = β60x2 + 1500x β 4000.
The domain of this function is the same as the domain of the original price-demand function, 1< x < 25 (x is in thousands of notebooks)
B) P(20) = 2000 (in thousands of dollars)
The profit is $2,000,000 when 20,000 notebooksare made and sold.
25
5000
Thousand dollars
Thousand notebooks
27