Chapter 1 Functions, Graphs, and Limits
Chapter 1
Functions, Graphs, and Limits
MA1103 Business Mathematics I
Semester I Year 2016/2017
SBM International Class
Lecturer: Dr. Rinovia Simanjuntak
1.1 Functions
2
Function
A function is a rule that assigns to each object in a
set A exactly one object in a set B.
The set A is called the domain of the function, and
the set of assigned objects in B is called the range.
3
Which One is a Function?
f
A
B
f
A
B
A
f
B
4
We represent a functional relationship by an equation
y f (x)
x and y are called variables: y is the dependent variable
and x is the independent variable.
Example.
y f ( x) x 2 4
Note that x and y can be substituted by other letters.
For example, the above function can be represented by
s t 4
2
5
Function which is Described as a Tabular Data
Table 1.1 Average Tuition and Fees for 4-Year Private Colleges
Academic Year
Ending in
1973
1978
1983
1988
1993
1998
2003
Period n
1
2
3
4
5
6
7
Tuition and
Fees
$1,898
$2,700
$4,639
$7,048
$10,448
$13,785
$18,273
6
We can describe this data as a function f defined by
the rule
average tuition and fees at the
f ( n)
beginning
of
the
n
th
5
year
period
Thus,
f (1) 1,898, f (2) 2,700, , f (7) 18,273
Noted that the domain of f is the set of integers
A {1,2,....,7}
7
Piecewise-defined function
A piecewise-defined function is a function that is
often defined using more than one formula, where
each individual formula describes the function on a
subset of the domain.
Example.
1
if x 1
f ( x) x 1
3x 2 1 if x 1
Find f(-1/2), f(1), and f(2).
8
Natural Domain
The natural domain of f is the domain of f to be the set
of all real numbers for which f(x) is defined.
There are two situations often need to be considered:
1) division by 0
2) the even root of a negative number
Examples.
Find the domain and range of each of these functions.
1. f ( x) 1 2
1 x
2. g (u ) 4 u 2
9
Functions Used in Economics
A demand function p=D(x) is a function that relates the unit
price p for a particular commodity to the number of units x
demanded by consumers at that price.
The total revenue is given by the product
R(x)=(number of items sold)(price per item)
=xp=xD(x)
If C(x) is the total cost of producing the x units, then the profit
is given by the function P(x)=R(x)-C(x)=xD(x)-C(x)
10
Example
Market research indicates that consumers will buy x
thousand units of a particular kind of coffee maker when
the unit price is p 0.27 x 51dollars. The cost of
producing the x thousand units is
C ( x) 2.23 x 2 3.5 x 85
thousand dollars
a. What are the revenue and profit functions, R(x) and
P(x), for this production process?
b. For what values of x is production of the coffee
makers profitable?
11
a. The demand function is D( x) 0.27 x 51 , so the revenue is
R( x) xD( x) 0.27 x 2 51x
thousand dollars, and the profit is (thousand dollars)
P( x) R ( x) C ( x)
0.27 x 2 51x (2.23 x 2 3.5 x 85)
2.5 x 2 47.5 x 85
b. Production is profitable when P(x)>0. We find that
P ( x ) 2.5 x 2 47.5 x 85
2.5( x 2 19 x 34)
2.5( x 2)( x 17 ) 0
Thus, production is profitable for 20, lim
and
A
0
x x k
lim
Example.
x2
Find lim
x 1 x 2 x 2
lim 1
x2
x2 / x2
1
x
lim
lim
0.5
2
x 1 x 2 x 2
x 1 / x 2 x / x 2 2 x 2 / x 2
lim 1 / x lim 1 / x lim 2 0 0 2
x
x
x
56
Procedure for Evaluating a Limit at Infinity of f(x)=p(x)/q(x)
Step 1. Divide each term in f(x) by the highest power xk that
appears in the denominator polynomial q(x).
f ( x) or lim f ( x) using algebraic
Step 2. Compute xlim
x
properties of limits and the reciprocal rules.
Example.
3x 4 8 x 2 2 x
lim
x
5x 4 1
57
Infinite Limits
If f(x) increases or decreases without bound as x→c, we
have lim f ( x) or
lim f ( x)
x c
Example. lim
x2
x c
x
( x 2) 2
From the figure, we
can guest that
x
lim
x 2 ( x 2) 2
58
1.6 One-sided Limits and
Continuity
59
One-Sided Limits
If f(x) approaches L as x tends toward c from the left
(xc), then
lim f ( x) M
x c
M is called the limit from the right (or right-hand
limit).
60
Example.
1 x 2 if x 2
For the function f ( x)
2 x 1 if x 2
evaluate the one-sided limits lim f ( x) and lim f ( x)
x 2
x 2
Since f ( x) 1 x 2 for x
Functions, Graphs, and Limits
MA1103 Business Mathematics I
Semester I Year 2016/2017
SBM International Class
Lecturer: Dr. Rinovia Simanjuntak
1.1 Functions
2
Function
A function is a rule that assigns to each object in a
set A exactly one object in a set B.
The set A is called the domain of the function, and
the set of assigned objects in B is called the range.
3
Which One is a Function?
f
A
B
f
A
B
A
f
B
4
We represent a functional relationship by an equation
y f (x)
x and y are called variables: y is the dependent variable
and x is the independent variable.
Example.
y f ( x) x 2 4
Note that x and y can be substituted by other letters.
For example, the above function can be represented by
s t 4
2
5
Function which is Described as a Tabular Data
Table 1.1 Average Tuition and Fees for 4-Year Private Colleges
Academic Year
Ending in
1973
1978
1983
1988
1993
1998
2003
Period n
1
2
3
4
5
6
7
Tuition and
Fees
$1,898
$2,700
$4,639
$7,048
$10,448
$13,785
$18,273
6
We can describe this data as a function f defined by
the rule
average tuition and fees at the
f ( n)
beginning
of
the
n
th
5
year
period
Thus,
f (1) 1,898, f (2) 2,700, , f (7) 18,273
Noted that the domain of f is the set of integers
A {1,2,....,7}
7
Piecewise-defined function
A piecewise-defined function is a function that is
often defined using more than one formula, where
each individual formula describes the function on a
subset of the domain.
Example.
1
if x 1
f ( x) x 1
3x 2 1 if x 1
Find f(-1/2), f(1), and f(2).
8
Natural Domain
The natural domain of f is the domain of f to be the set
of all real numbers for which f(x) is defined.
There are two situations often need to be considered:
1) division by 0
2) the even root of a negative number
Examples.
Find the domain and range of each of these functions.
1. f ( x) 1 2
1 x
2. g (u ) 4 u 2
9
Functions Used in Economics
A demand function p=D(x) is a function that relates the unit
price p for a particular commodity to the number of units x
demanded by consumers at that price.
The total revenue is given by the product
R(x)=(number of items sold)(price per item)
=xp=xD(x)
If C(x) is the total cost of producing the x units, then the profit
is given by the function P(x)=R(x)-C(x)=xD(x)-C(x)
10
Example
Market research indicates that consumers will buy x
thousand units of a particular kind of coffee maker when
the unit price is p 0.27 x 51dollars. The cost of
producing the x thousand units is
C ( x) 2.23 x 2 3.5 x 85
thousand dollars
a. What are the revenue and profit functions, R(x) and
P(x), for this production process?
b. For what values of x is production of the coffee
makers profitable?
11
a. The demand function is D( x) 0.27 x 51 , so the revenue is
R( x) xD( x) 0.27 x 2 51x
thousand dollars, and the profit is (thousand dollars)
P( x) R ( x) C ( x)
0.27 x 2 51x (2.23 x 2 3.5 x 85)
2.5 x 2 47.5 x 85
b. Production is profitable when P(x)>0. We find that
P ( x ) 2.5 x 2 47.5 x 85
2.5( x 2 19 x 34)
2.5( x 2)( x 17 ) 0
Thus, production is profitable for 20, lim
and
A
0
x x k
lim
Example.
x2
Find lim
x 1 x 2 x 2
lim 1
x2
x2 / x2
1
x
lim
lim
0.5
2
x 1 x 2 x 2
x 1 / x 2 x / x 2 2 x 2 / x 2
lim 1 / x lim 1 / x lim 2 0 0 2
x
x
x
56
Procedure for Evaluating a Limit at Infinity of f(x)=p(x)/q(x)
Step 1. Divide each term in f(x) by the highest power xk that
appears in the denominator polynomial q(x).
f ( x) or lim f ( x) using algebraic
Step 2. Compute xlim
x
properties of limits and the reciprocal rules.
Example.
3x 4 8 x 2 2 x
lim
x
5x 4 1
57
Infinite Limits
If f(x) increases or decreases without bound as x→c, we
have lim f ( x) or
lim f ( x)
x c
Example. lim
x2
x c
x
( x 2) 2
From the figure, we
can guest that
x
lim
x 2 ( x 2) 2
58
1.6 One-sided Limits and
Continuity
59
One-Sided Limits
If f(x) approaches L as x tends toward c from the left
(xc), then
lim f ( x) M
x c
M is called the limit from the right (or right-hand
limit).
60
Example.
1 x 2 if x 2
For the function f ( x)
2 x 1 if x 2
evaluate the one-sided limits lim f ( x) and lim f ( x)
x 2
x 2
Since f ( x) 1 x 2 for x