Chapter 1Functions, Graphs, and Limits

MA1103 Business Mathematics I

Semester I Year 2016/2017

SBM International Class

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 0 2) 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 particular commodity to the number of units xdemanded 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 profitis 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 dollars. The cost of producing the x thousand units is

5127.0 xp

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?

12

a. The demand function is , so the revenue is

thousand dollars, and the profit is (thousand dollars)

5127.0)( xxD

xxxxDxR 5127.0)()( 2

855.475.2

)855.323.2(5127.0

)()()(

2

22

xx

xxxx

xCxRxP

b. Production is profitable when P(x)>0. We find that

0)17)(2(5.2

)3419(5.2

855.475.2)(

2

2

xx

xx

xxxP

Thus, production is profitable for 2<x<17.

Composition of Functions

13

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).

1)( xxg

1)( 3 uuf

2331)1())(( 233 xxxxxgf

Example.

Find the composition function f(g(x)), where and

Solution.

Replace u by x+1 in the formula for f(u) to get

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

14

GraphThe graph of a function f consists of all points (x,y) where xis 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

15

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 ,

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

Note: Sometimes finding x intercepts may be difficult.

Following 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).

)0(0 fy

16

Parabolas

Parabolas: The graph of as long as 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

CBxAxy 2

17

A

Bx

2

18

Example 6

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?

Solution:

The revenue function R(x)=x(60-x) hundred dollars. Note that R(x) ≥0

only for 0≤x≤60. The revenue function can be rewritten as

xxxR 60)( 2

which is a parabola that opens downward (Since A=-1<0) and has its

high point (vertex) at 30)1(2

60

2

A

Bx

Thus, revenue is maximized when x=30 hundred units are produced,

and the corresponding maximum revenue is R(30)=900 hundred

dollars.

Intersections of Graphs

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

19

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 , where n is a real number.

A polynomial function: A function of the form

where n is a nonnegative integer and are constants.

If , the integer n is called the degree of the polynomial.

A rational function: A quotient of two polynomials p(x) and q(x).

01

1

1)( axaxaxaxp n

n

n

n

nxxf )(

20

naaa ,,, 10

0na

)(

)(

xq

xp

The Vertical Line Test

21

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

22

Linear Functions

bmxy

23

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

where m and b are constants.

The Slope of a Line

The slope of the non-vertical line passing through the points

and is given by the formula),( 11 yx

12

12

in x change

yin changeSlope

xx

yy

x

y

24

),( 22 yx

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

The point-slope form: The equation is an equation of the line that passes through the point

and has slope equal to m.

bmxy

)( 00 xxmyy

25

),( 00 yx

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

26

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

27

2m1L

1L 2L 21 mm

1L 2L1

2

1

mm

28

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

4Lm

1L3

8

3

4)1(

3

44 xyxy

2

9

4

3

)2(4

33

xy

xy2L

1.4 Functional Models

29

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.

30

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.

31

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.

32

Solution:

We denote x and y as the lengths of the sides of the picnic area.

Expressing the number of yards F of required fencing in terms of

these two variables, we get . Using the fact that the area

is to be 5,000 square yards that is

yxF 2

xyxy

5000000,5

and substitute the resulting expression for y into the formula for F to

getx

xx

xxF100005000

2)(

33

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?

34

Solution:

a. As we know, Profit=(number of cassettes sold)(profit per

cassette)

Let p denote the price at which each cassette will be sold

and let P(p) be the corresponding monthly profit.

Number of cassettes sold

=4000-400(number of $1 increases)

=4000-400(p-5)=6000-400p

Profit per cassette=p-2

The total profit is 120006800400

)2)(4006000()(

2

pp

pppP

35

b. The graph of P(p) is the downward opening parabola

shown in the bottom figure. Profit is maximized at the

value of p that corresponds to the vertex of the parabola.

We know

Thus, profit is maximized when the manufacturer charges

$8.50 for each cassette, and the maximum monthly profit

is

5.8)400(2

6800

2

A

Bp

16900$12000)5.8(6800)5.8(400)5.8( 2

max PP

Market Equilibrium

)()( eee xSxDp

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:

Shortage: D(x)>S(x)

Surplus: S(x)>D(x)

37

Market research indicates that manufacturers will supply x

units 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

xxDxxS 6174)( 14)( 2

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.

38

Solution:

a. Market equilibrium occurs when S(x)=D(x), we have

16or 10

0)16)(10(

6174142

x

xx

xx

Only positive values are meaningful, 114)10(6174)10( Dpe

Break-Even Analysis

39

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.

40

A manufacturer can sell a certain product for $110 per unit.

Total cost consists of a fixed overhead of $7500 plus

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 100 units are

sold?

c.How many units must be sold for the manufacturer to

realize a profit of $1250?

Solution:

If x is the number of units manufactured and sold, the total

revenue is given by R(x)=110x and the total cost by

C(x)=7500+60x

41

a. To find the break-even point, set R(x) equal to C(x) and solve

110x=7500+60x, so that x=150.

It follows that the manufacturer will have to sell 150 units to break

even.

b. The profit P(x) is revenue minus cost. Hence,

P(x)=R(x)-C(x)=110x-(7500+60x)=50x-7500

The profit from the sale of 100 units is P(100)=-2500

It follows that the manufacturer will lose $2500 if 100 units are

sold.

c. We set the formula for profit P(x) equal to 1250 and solve for x,

we have P(x)=1250, x=175. That is 175 units must be sold to

generate the desired profit.

42

A certain car rental agency charges $25 plus 60 cents per

mile. A second agency charge $30 plus 50 cents per mile.

Which agency offers the better deal?

Solution:

Suppose a car is to be driven x miles, then the first agency

will charge dollars and the second will charge

. So that x=50.

For shorter distances, the first agency offers the better deal,

and for longer distances, the second agency is better.

xxC 60.025)(1

xxC 50.030)(2

1.5 Limits

43

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? 44

96068

3206368)(

2

2

xx

xxxC

45

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 7)(lim80

xCx

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

46

Lxfcx

)(lim

Example

Use a table to estimate the limit

47

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

48

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

49

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

50

)](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

51

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

52

Find (a) (b)

a. Apply the properties of limits to obtain

98)1(4)1(38limlim4lim3)843(lim 3

11

3

1

3

1

xxxxxxxx

b. Since , you can use the quotient rule for

limits to get

0)2(lim0

xx

420

80

2limlim

8limlim3

2

83lim

00

0

3

0

3

0

xx

xx

x x

x

x

x

Limits of Polynomials and Rational Functions

53

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

54

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

a. 21

2

2

1lim

)2)(1(

)1)(1(lim

23

1lim

112

2

1

x

x

xx

xx

xx

x

xxx

b. 2

1

1

1lim

1)1(

1lim

1)1(

11lim

1

1lim

1111

xxx

x

xx

xx

x

x

xxxx

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

55

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

56

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

57

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

58

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

59

One-Sided Limits

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

Lxfcx

)(lim

Mxfcx

)(lim

60

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

61

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

62

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

63

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

64

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

65

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

66

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.

67

Continuity

A continuous function is one whose graph can be drawn

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

68

A “hole “ at x=c

69

A “gap” at x=c

70

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

71

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

72

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

73

Example.

Show that the rational function is

continuous at x=3.2

1)(

x

xxf

Note that f(3)=(3+1)/(3-2)=4,

since , you will find that 0)2(lim3

xx

)3(41

4

)2(lim

)1(lim

2

1lim)(lim

3

3

33f

x

x

x

xxf

x

x

xx

as required for f(x) to be continuous at x=3, since the

three criteria for continuity are satisfied.

74

Example.

Determine where the function below is not continuous.

Rational functions are continuous everywhere except

where we have division by zero.

The function given will not be continuous at t=-3

and t=5.

75

Example.

Discuss the continuity of each of the following functions

1 if 2

1 if 1)( .

1

1)( .

1)( .

2

xx

xxxhc

x

xxgb

xxfa

76

Example.For what value of the constant A is the following function

continuous for all real x?

1 xif 43

1 if 5)(

2 xx

xAxxf

Since Ax+5 and are both polynomials, it follows that

f(x) will be continuous everywhere except possibly at x=1 .

According to the three criteria for continuity, we have

432 xx

This means that f is continuous for all x only when A=-3

3)1(25)1()(lim)(lim11

AfAfxfxfxx

77

Example.

Find numbers a and b so that the following function is

continuous everywhere.

1 if

11 if

1 if

)( 2

xbx

xbax

-xax

xf

Since the “parts” f are polynomials, we only need to

choose a and b so that f is continuous at x=-1 and 1.

At x=-1 121)1()(lim)(lim11

babaafxfxfxx

At x=1 121)1()(lim)(lim11

babbafxfxfxx

We have a=-1/3 and b=1/3 for f is continuous everywhere

78

Continuity on an Interval

A function f(x) is said to be continuous on an open interval

a<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, and

is continuous on [-1,1]

)()(lim afxfax

)()(lim bfxfbx

21)( xxf

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Example.

Discuss the continuity of the function on the open

interval -2<x<3 and on the closed interval -2≤x≤33

2)(

x

xxf

The rational function f(x)is continuous for all x except x=3.

Therefore, it is continuous on the open interval -2<x<3 but not

on the closed interval -2≤x≤3, since it is discontinuous at the

endpoint 3 (where its denominator is zero).

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Summary

Function:

Domain and range of a function

Composition of function f(g(x))

Graph of a function:

x and y intercepts,

Piecewise-defined function, power function

Polynomial, rational function, vertical line test

Linear function:

Slope, slope-intercept formula, point-slope formula

Parallel and perpendicular lines

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Function Models:

Market equilibrium: law of supply and demand

Shortage and surplus, break-even analysis

Limits:

Calculation of limits, limits of polynomial and

rational function

Limits at infinity: limits at the infinity (Reciprocal power

Rules), infinite limit

One sided limit, existence of limit

Continuity of f(x) at x=c:

Continuity of polynomials and rational function

Lxfcx

)(lim