Chapter 1Functions, Graphs, and Limits

MA1103 Business Mathematics I

Semester I Year 2019/2020

SBM International Class K-10

Lecturer: Dr. Rinovia Simanjuntak

1.1 Functions

2

Function

3

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.

Which One is a Function?

fff

A B

AB

f

A B f

4

)(xfy =

5

We represent a functional relationship by an equation

x and y are called variables: y is the dependent variableand x is the independent variable.

Example.

Note that x and y can be substituted by other letters. For example, the above function can be represented by

4)( 2 +== xxfy

42 += ts

Function which is Described as a Tabular Data

Academic Year Tuition and

Ending in Period n Fees

1973 1 $1,898

1978 2 $2,700

1983 3 $4,639

1988 4 $7,048

1993 5 $10,448

1998 6 $13,785

2003 7 $18,273

6

Table 1.1 Average Tuition and Fees for 4-Year Private Colleges

=

periodyear -5th theof beginning

at the fees and tuition average)(

nnf

7

We can describe this data as a function f defined by the rule

Thus,

Noted that the domain of f is the set of integers

273,18)7(,,700,2)2(,898,1)1( === fff

}7,....,2,1{=A

Piecewise-defined function

8

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.

Find 𝑓(−1/2), 𝑓(1), and 𝑓(2).

+

−=

1 xif 13

1 xif 1

1

)(2x

xxf

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.

Examples.

Find the domain and range of each of these functions.

1.

2. 9

There are two situations often need to be considered:1) division by 02) the even root of a negative number

21

1)(

xxf

−=

4 2)( += uug

Functions Used in Economics

A demand function 𝑝 = 𝐷(𝑥) is a function that relates the unit price 𝑝 for a commodity to the number of units 𝑥demanded by consumers at that price.

The total revenue is given by the product𝑅 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

11

ExampleMarket research indicates that consumers will buy 𝑥thousand units of a particular kind of coffee maker when the unit price is 𝑝 = −0.27𝑥 + 51 dollars. The cost of producing the 𝑥 thousand units is

𝐶 𝑥 = 2.23 𝑥2 + 3.5𝑥 + 85

855.323.2)( 2 ++= xxxC

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?

Composition of Functions

12

Given functions 𝑓(𝑢) and 𝑔(𝑥), the composition 𝑓(𝑔(𝑥)) is the function of 𝑥 formed by substituting 𝑢 = 𝑔(𝑥) for 𝑢 in the formula for 𝑓(𝑢).

Example.

Find the composition function 𝑓(𝑔(𝑥)), where 𝑓 𝑢 = 𝑢3 + 1and 𝑔 𝑥 = 𝑥 + 1.

Question: How about 𝑔(𝑓(𝑥))?

Note: In general, 𝑓(𝑔(𝑥)) and 𝑔(𝑓(𝑥)) are not the same.

1.2 The Graph of a Function

13

GraphThe graph of a function 𝑓 consists of all points (𝑥, 𝑦) where 𝑥 is in the domain of 𝑓 and 𝑦 = 𝑓(𝑥), that is, all points of the form (𝑥, 𝑓(𝑥)).

Rectangular coordinate system, horizontal axis, vertical axis.

2)( 2 ++−= xxxf

x -3 -2 -1 0 1 2 3 4

f(x) -10 -4 0 2 2 0 -4 -10

14

Intercepts

𝑥 intercept: points where a graph crosses the x axis.

𝑦 intercept: a point where the graph crosses the y axis.

How to find the 𝒙 and 𝒚 intercepts:

• The only possible 𝑦 intercept for a function is 𝑦0 = 𝑓(0).

• To find any 𝑥 intercept of 𝑦 = 𝑓(𝑥), set 𝑦 = 0 and solve for 𝑥.

Note: Sometimes finding 𝑥 intercepts may be difficult.

In the aforementioned example, the 𝑦 intercept is 𝑓(0) = 2. To find the 𝑥 intercepts, solve the equation 𝑓(𝑥) = 0, we have 𝑥 = −1 and 2. Thus, the 𝑥 intercepts are (−1,0) and (2,0). 15

Parabolas

Parabolas: The graph of 𝑦 = 𝐴𝑥2 + 𝐵𝑥 + 𝐶, 𝐴 ≠ 0.

All parabolas have a “U shape” and the parabola opens up if 𝐴 > 0 and down if 𝐴 < 0.

The “peak” or “valley” of the parabola is called its vertex,

and it always occurs where 𝑥 =−𝐵

2𝐴.

16

Example

A manufacturer determines that when 𝑥 hundred units of a particular commodity are produced, they can all be sold for a unit price given by the demand function 𝑝 = 60 − 𝑥 dollars. At what level of production is revenue maximized? What is the maximum revenue?

17

Intersections of Graphs

Sometimes it is necessary to determine when two functions are equal.

18

For example, an economist may wish to compute the market price at which the consumer demand for a commodity will be equal to supply.

Power, Polynomial, and Rational Functions

A power function: A function of the form 𝑓 𝑥 = 𝑥𝑛, where 𝑛 is a real number.

A polynomial function: A function of the form 𝑝 𝑥 = 𝑎𝑛𝑥

𝑛 + 𝑎𝑛−1𝑥𝑛−1 +⋯+ 𝑎1𝑥 + 𝑎0

where 𝑛 is a nonnegative integer and 𝑎0, 𝑎1, … , 𝑎𝑛 are constants.

If 𝑎𝑛 ≠ 0, the integer 𝑛 is called the degree of the polynomial.

A rational function: A quotient 𝑝(𝑥)

𝑞(𝑥)of two polynomials 𝑝(𝑥) and 𝑞(𝑥).

19

The Vertical Line Test

20

A curve is the graph of a function if and only if no vertical line intersects the curve more than once.

1.3 Linear Functions

21

Linear Functions

22

A linear function is a function that changes at a constant rate with respect to its independent variable.

The graph of a linear function is a straight line.

The equation of a linear function can be written in the form

𝑦 = 𝑚𝑥 + 𝑏

where 𝑚 and 𝑏 are constants.

The Slope of a Line

The slope of the non-vertical line passing through the points (𝑥1, 𝑦1) and (𝑥2, 𝑦2) is given by the formula

slope =change in 𝑦

change in 𝑥=∆𝑦

∆𝑥=𝑦2 − 𝑦1𝑥2 − 𝑥1

23

Equation of a Line The slope-intercept form: The equation 𝑦 = 𝑚𝑥 + 𝑏 is the equation of a line whose slope is 𝑚 and whose 𝑦 intercept is (0, 𝑏).

The point-slope form: The equation 𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0)is an equation of the line that passes through the point (𝑥0, 𝑦0) and has slope equal to 𝑚.

24

The slope-intercept form is 3

1

)05.1(

)5.00(−=

−−

+=m

2

1

3

1−−= xy

The point-slope form that passes through the point (−1.5,0) is

)5.1(3

10 +−=− xy

25

Table 1.2 lists the percentage of the labour force that was unemployed during the decade 1991-2000. Plot a graph with the time (years after 1991) on the x axis and percentage of unemployment on the y axis. Do the points follow a clear pattern? Based on these data, what would you expect the percentage of unemployment to be in the year 2005?

Number of Years Percentage of

Year from 1991 Unemployed

1991 0 6.8

1992 1 7.5

1993 2 6.9

1994 3 6.1

1995 4 5.6

1996 5 5.4

1997 6 4.9

1998 7 4.5

1999 8 4.2

2000 9 4.0

Table 1.2 Percentage of Civilian Unemployment

Parallel and Perpendicular Lines

Let and be the slope of the non-vertical lines

and . Then

and are parallel if and only if

and are perpendicular if and only if

1m

2L

26

2m1L

1L 2L 21 mm =

1L 2L1

2

1

mm −=

27

Let L be the line 4x+3y=3

a. Find the equation of a line parallel to L through P(-1,4).

b. Find the equation of a line perpendicular to L through Q(2,-3).

1L

2L

Solution:

By rewriting the equation 4x+3y=3 in the slope-intercept form

, we see that L has slope

a. Any line parallel to L must also have slope -4/3. The required line

contains P(-1,4), we have

b. A line perpendicular to L must have slope m=3/4. Since the

required line contains Q(2,-3), we have

13

4+−= xy

3

4−=Lm

1L3

8

3

4)1(

3

44 +−=+−=− xyxy

2

9

4

3

)2(4

33

−=

−=+

xy

xy2L

1.4 Functional Models

28

Functional Models

To analyze a real world problem, a common procedure is to make assumptions about the problem that simplify it enough to allow a mathematical description. This process is called mathematical modelling and the modified problem based on the simplifying assumptions is called a mathematical model.

29

Real-world

problem

Testing

Interpretation

Mathematical

model

adjustments

PredictionAnalysis

Formulation

Elimination of Variables

In next example, the quantity you are seeking is expressed most naturally in term of two variables. We will have to eliminate one of these variables before you can write the quantity as a function of a single variable.

30

Example

The highway department is planning to build a picnic area for motorists along a major highway. It is to be rectangular with an area of 5,000 square yards and is to be fenced off on the three sides not adjacent to the highway. Express the number of yards of fencing required as a function of the length of the unfenced side.

31

32

Modelling in Business and Economics

A manufacturer can produce blank videotapes at a cost of $2 per cassette. The cassettes have been selling for $5 a piece. Consumers have been buying 4000 cassettes a month. The manufacturer is planning to raise the price of the cassettes and estimates that for each $1 increase in the price, 400 fewer cassettes will be sold each month.

a. Express the manufacturer’s monthly profit as a function of the price at which the cassettes are sold.

b. Sketch the graph of the profit function. What price corresponds to maximum profit? What is the maximum profit?

33

Market Equilibrium

The law of supply and demand: In a competitive market environment, supply tends to equal demand, and when this occurs, the market is said to be in equilibrium.

The demand function:

𝑝 = 𝐷(𝑥)

The supply function:

𝑝 = 𝑆(𝑥)

The equilibrium price:

𝑝𝑒 = 𝐷 𝑥𝑒 = 𝑆(𝑥𝑒)

Shortage: 𝐷(𝑥) > 𝑆(𝑥)

Surplus: 𝑆(𝑥) > 𝐷(𝑥)

Example

Market research indicates that manufacturers will supply 𝑥units of a particular commodity to the marketplace when the price is 𝑝 = 𝑆(𝑥) dollars per unit and that the same number of units will be demanded by consumers when the price is 𝑝 =𝐷(𝑥) dollars per unit, where the supply and demand functions are given by

𝑆 𝑥 = 𝑥2 + 14 𝐷 𝑥 = 174 − 6𝑥

a. At what level of production 𝑥 and unit price 𝑝 is market equilibrium achieved?

b. Sketch the supply and demand curves, 𝑝 = 𝑆(𝑥) and 𝑝 =𝐷(𝑥), on the same graph and interpret.

35

36

Break-Even Analysis

37

At low levels of production, the manufacturer suffers a

loss. At higher levels of production, however, the total

revenue curve is the higher one and the manufacturer

realizes a profit.

Break-even point: The total revenue equals total cost.

Example

A manufacturer can sell a certain product for $110 per unit. Total cost consists of a fixed overhead of $7500plus production costs of $60 per unit.

a. How many units must the manufacturer sell to break even?

b. What is the manufacturer’s profit or loss if 100units are sold?

c. How many units must be sold for the manufacturer to realize a profit of $1250?

38

Example (2)

A certain car rental agency charges $25 plus 60cents per mile. A second agency charge $30plus 50 cents per mile.

Which agency offers the better deal?

39

1.5 Limits

40

Illustration of Limit

The limit process involves examining the behaviour of a function f(x) as x approaches a number c that may or may not be in the domain of f.

Illustration.

Consider a manager who determines that when x percent of her company’s plant capacity is being used, the total cost is

hundred thousand dollars. The company has a policy of rotating maintenance in such a way that no more than 80% of capacity is ever in use at any one time. What cost should the manager expect when the plant is operating at full permissible capacity? 41

96068

3206368)(

2

2

−−

−−=

xx

xxxC

42

It may seem that we can answer this question by simply

evaluating C(80), but attempting this evaluation results in

the meaningless fraction 0/0.

However, it is still possible to evaluate C(x) for values of

x that approach 80 from the left (x<80) and the right

(x>80), as indicated in this table:

x approaches 80 from the left → ←x approaches 80 from the right

x 79.8 79.99 79.999 80 80.0001 80.001 80.04

C(x) 6.99782 6.99989 6.99999 7.000001 7.00001 7.00043

The values of C(x) displayed on the lower line of this table

suggest that C(x) approaches the number 7 as x gets closer

and closer to 80. The functional behavior in this example

can be describe by lim𝑥→80 𝐶 𝑥 = 7.

Limits

If 𝑓(𝑥) gets closer and closer to a number 𝐿 as 𝑥 gets closer and closer to c from both sides, then 𝐿 is the limit of 𝑓(𝑥) as 𝑥 approaches 𝑐. The behaviour is expressed by writing

lim𝑥→𝑐

𝑓(𝑥) = 𝐿

43

Example

Use a table to estimate the limit

44

1

1)(

−

−=

x

xxfLet and compute f(x) for a succession of values

of x approaching 1 from the left and from the right.

1

1lim

1 −

−

→ x

x

x

x→ 1 ← x

x 0.99 0.999 0.9999 1 1.00001 1.0001 1.001

f(x) 0.50126 0.50013 0.50001 0.499999 0.49999 0.49988

The table suggest that f(x) approaches 0.5 as x approaches

1. That is 5.0

1

1lim

1=

−

−

→ x

x

x

45

Three functions for which

It is important to remember that limits describe the behavior of a function near a particular point, not necessarily at the point itself.

4)(lim3

=→

xfx

46

The figure below shows that the graph of two functions

that do not have a limit as x approaches 2.

Figure (a): The limit does not exist;

Figure (b): The function has no finite limit as x

approaches 2. Such so-called infinite limits will be

discussed later.

Properties of Limits

thenexist, )(limand)(lim If xgxfcxcx →→

)(lim)(lim)]()([lim xgxfxgxfcxcxcx →→→

+=+

)(lim)(lim)]()([lim xgxfxgxfcxcxcx →→→

−=−

constant any for )(lim)(lim kxfkxkfcxcx →→

=

47

)](lim)][(lim[)]()([lim xgxfxgxfcxcxcx →→→

=

0)(lim if )(lim

)(lim]

)(

)([lim =

→

→

→

→xg

xg

xf

xg

xf

cx

cx

cx

cx

exists )](lim[ if )](lim[)]([lim p

cx

p

cx

p

cxxfxfxf

→→→=

48

For any constant k,

That is, the limit of a constant is the constant itself, and

the limit of f(x)=x as x approaches c is c.

cxkkcx

==→→ cx

lim and lim

Examples

)843(lim 3

1+−

−→xx

x 2

83lim

3

0 −

−

→ x

x

x

49

Find (a) (b)

Limits of Polynomials and Rational Functions

50

If p(x) and q(x) are polynomials, then

and

)()(lim cpxpcx

=→

0)( if )(

)(

)(

)(lim =→

cqcq

cp

xq

xp

cx

Example.

Find 2

1lim

2 −

+

→ x

x

x

The quotient rule for limits does not apply in this case since the limit of

the denominator is 0 and the limit of the numerator is 3.

Indeterminate Form

51

If and , then is said to be

indeterminate. The term indeterminate is used since the limit

may or may not exist.

Examples.

(a) Find (b) Find

0)(lim =→

xfcx

0)(lim =→

xgcx )(

)(lim

xg

xf

cx→

23

1lim

2

2

1 +−

−

→ xx

x

x 1

1lim

1 −

−

→ x

x

x

Limits Involving Infinity

52

Limits at Infinity

If the value of the function f(x) approach the number L as x increases

without bound, we write

Similarly, we write

when the functional values f(x) approach the number M as x decreases

without bound.

Lxfx

=+→

)(lim

Mxfx

=−→

)(lim

53

Reciprocal Power Rules

For constants A and k, with k>0, 0lim and 0lim ==−→+→ kxkx x

A

x

A

Example.

Find2

2

21lim

xx

x

x +++→

5.0200

1

2lim/1lim/1lim

1lim

/2//1

/lim

21lim

22222

22

2

2

=++

=++

=++

=++

+→+→+→

+→

+→+→

xxx

x

xx xxxxxxx

xx

xx

x

54

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).

Step 2. Compute or using algebraic

properties of limits and the reciprocal rules.

)(lim xfx +→

)(lim xfx −→

Example.

15

283lim

4

24

+

+−

+→ x

xxx

x

55

Infinite Limits

If f(x) increases or decreases without bound as x→c, we

have )(limor )(lim xf xfcxcx

−=+=→→

Example. 22 )2(lim

−→ x

x

x

From the figure, we

can guest that

+=−→ 22 )2(

limx

x

x

1.6 One-sided Limits and Continuity

56

One-Sided Limits

If f(x) approaches L as x tends toward c from the left (x<c), we write

Lxfcx

=−→

)(lim

Mxfcx

=+→

)(lim

57

where L is called the limit from the left (or left-hand

limit)

Likewise if f(x) approaches M as x tends toward c

from the right (x>c), then

M is called the limit from the right (or right-hand

limit).

Example.

For the function

evaluate the one-sided limits and

58

+

−=

2 if 12

2 if 1)(

2

xx

xxxf

)(lim2

xfx −→

)(lim2

xfx +→

Since for x<2, we have 21)( xxf −=

3)1(lim)(lim 2

22−=−=

−− →→xxf

xx

Similarly, f(x)=2x+1 if x≥2, so

5)12(lim)(lim22

=+=++ →→

xxfxx

59

Existence of a Limit

The two-sided limit exists if and only if the two

one-sided limits and exist and are

equal, and then

)(lim xfcx→

)(lim xfcx −→

)(lim xfcx +→

)(lim)(lim)(lim xfxfxfcxcxcx +− →→→

==

Recall.

Find 2

1lim

2 −

+

→ x

x

x

60

At x=1: ( )1

lim 0x

f x−→

=

( )1

lim 1x

f x+→

=

( )1 1f =

Left-hand limit

Right-hand limit

value of the function

does not exist!

Since the left and right hand

limits are not equal.

)(lim1

xfx→

61

At x=2: Left-hand limit

Right-hand limit

value of the function

( )2

lim 1x

f x−→

=

( )2

lim 1x

f x+→

=

( )2 2f =

does exist!

Since the left and right

hand limits are equal.

However, the limit is not

equal to the value of

function.

)(lim2

xfx→

62

At x=3: Left-hand limit

Right-hand limit

value of the function

( )3

lim 2x

f x−→

=

( )3

lim 2x

f x+→

=

( )3 2f =

does exist!

Since the left and right

hand limits are equal,

and the limit is equal

to the value of

function.

)(lim3

xfx→

63

Non-existent One-sided Limits

A simple example is provided by the function

)/1sin()( xxf =

As x approaches 0 from either the left or the right, f(x)

oscillates between -1 and 1 infinitely often. Thus neither

one-sided limit at 0 exists.

64

Continuity

A continuous function is one whose graph can be drawn

without the “pen” leaving the paper. (no holes or gaps )

65

A “hole “ at x=c

66

A “gap” at x=c

67

What properties will guarantee that f(x) does not have a “hole”

or “gap” at x=c?

A function f is continuous at c if all three of these conditions

are satisfied:

a.

b.

c.

If f(x) is not continuous at c, it is said to have a discontinuity

there.

exists )(lim xfcx→

)()(lim cfxfcx

=→

defined is )(cf

68

f(x) is continuous at

x=3 because the left

and right hand limits

exist and equal to f(3).

At x=1:

At x=2:

At x=3:

)(lim)(lim11

xfxfxx −+ →→

)2()(lim)(lim22

fxfxfxx

=−+ →→

)3()(lim)(lim33

fxfxfxx

==−+ →→

Discontinuous

Discontinuous

Continuous

69

Continuity of Polynomials and

Rational Functions

If p(x) and q(x) are polynomials, then

)()(lim cpxpcx

=→

0)( if )(

)(

)(

)(lim =→

cqcq

cp

xq

xp

cx

A polynomial or a rational function is continuous

wherever it is defined

Example

1. Show that the rational function 𝑓 𝑥 =𝑥+1

𝑥−2is

continuous at 𝑥 = 3.

2. Determine where the function below is not continuous.

ℎ 𝑡 =4𝑡 + 10

𝑡2 − 2𝑡 − 15

70

Example (2)

Discuss the continuity of each of the following functions

71

−

+=

+

−==

1 if 2

1 if 1)( .

1

1)( .

1)( .

2

xx

xxxhc

x

xxgb

xxfa

Example (3)

1. For what value of the constant A is the following function continuous for all real x?

2. Find numbers a and b so that the following function is continuous everywhere.

72

+−

+=

1 xif 43

1 if 5)(

2 xx

xAxxf

−−+

=

1 if

11 if

1 if

)( 2

xbx

xbax

-xax

xf

Continuity on an Interval

A function 𝑓(𝑥) is said to be continuous on an open interval𝑎 < 𝑥 < 𝑏 if it is continuous at each point 𝑥 = 𝑐 in that interval.

f is continuous on closed interval 𝑎 ≤ 𝑥 ≤ 𝑏, if it continuous on the open interval 𝑎 < 𝑥 < 𝑏, lim𝑥→𝑎+ 𝑓(𝑥) = 𝑓(𝑎) and lim𝑥→𝑏−

𝑓(𝑥) = 𝑓 𝑏 .

Example.

1. 𝑓 𝑥 = 1 − 𝑥2

2. Discuss the continuity of the function 𝑔 𝑥 =𝑥+2

𝑥−3on the open interval −2 < 𝑥 < 3 and on the closed interval −2 ≤ 𝑥 ≤ 3.

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