biz - quatitative.managment.method chapter.04

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Chapter IVChapter IV

Relations and FunctionsRelations and Functions

22

Ordered Pairs: In writing a set {a, b}, we do not care

about the order in which the elements aand b appear, because, by definition {a, b}= {b, a}. The pair of elements a and b is inthis case an unordered pair.

When the ordering of a and b does carrya significance, we can write two differentordered pairs denoted by (a, b) and (b, a),which have the property that (a, b) (b, a)unless a = b.


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Similarly, we can distinguish between orderedand unordered triples, quadruples, quintuples,and so forth.

Ordered pairs, triples, etc., collectively can becalled ordered sets.

Ordered pairs, like other objects, can beelements of set S.

Consider the rectangular (cartesian coordinate

plane) where an x axis and a y axis crosseach other at a right angle, dividing the planeinto four quadrants.

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The xy plane is an infinite set of points, each ofwhich represents an ordered pair whose firstelement is an x value and the second element a yvalue. Clearly, the point labeled (4, 2) is differentfrom the point (2, 4); thus ordering is significanthere.

Suppose, from two given sets, x = {1, 2} and y ={3, 4}, we wish to form all the possible orderedpairs with the first element taken from set x and thesecond element taken from set y. The result will, ofcourse, be the set of four ordered pairs (1,3), (1,4), (2, 3), and (2, 4). This set is called the cartesianproduct or direct product of the set x and y and isdenoted by x X y (x cross y).


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While x and y are sets of numbers, thecartesian product turns out to be a set ofordered pairs. By enumeration, or by

description, we may express the cartesianproduct alternatively as

x X y = {(1, 3), (1, 4), (2, 3), (2, 4)}

or x X y = {(a, b) | a x and b y}

Let x and y include all the real numbers. Thenthe cartesian product

x X y = {(a, b) | a R and b R} .(1)

will represent the set of all ordered pairs withreal-valued elements.

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Each ordered pair corresponds to unique pointin the cartesian coordinate plane and,conversely, each point in the coordinate planealso corresponds to a unique ordered pair in theset x X y.

In view of this double uniqueness, a one-to-onecorrespondence is said to exist between the set

of ordered pairs in the cartesian product andthe set of points in the rectangular coordinateplane.

Another way of expressing the set x X y in (1) isto write it directly as R X R; this is alsocommonly denoted by R2.


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We may also define the cartesian product ofthree sets x, y, and z as follows:

x X y X z = {(a, b, c)| a x, b y, c z}Which is a set of ordered triples.

If the sets x, y and z each consist of all the realnumbers, the cartesian product will correspondto the set of points in a three dimensionalspace. This may be denoted by R X R X R, or

more simply, R3

. In this way, we can have R2, or R3,, or Rn.

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Relation:Relation:

Since any ordered pair associates a yvalue with an x value, any collection ofordered pairs any subset of thecartesian product (1) will constitute a

relation between y and x.

Given an x value, one or more y valueswill be specified by that relation.


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Example 1:

The set {(x, y) | y = 2x} is a set of ordered pairs

including for example, (1, 2), (0, 0), and (-1, -2). It constitutes a relation, and its graphicalcounterpart is the set of points lying on thestraight line y = 2x (see Fig. 1).

Example 2:

The set {(x, y) | y x}, which consists of such

ordered pairs as (1, 0), (1, 1), and (1, -4),constitutes another relation (see Fig. 1).

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When the x value is given, it may not be alwayspossible to determine a unique y value from arelation.

In Example 2, the three exemplary orderedpairs show that if x = 1, y can take variousvalues, such as 0, 1, or -4, and yet in each casesatisfy the stated relation.

A relation may be such that for each x valuethere exists only one corresponding y value.

The relation in Example 1 indicates that y issaid to be a function of x, and this is denoted byy = f(x).


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x

y x = a y = 2x y = x

a

y x

O

Fig. 1

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00x1

x2

y = f (x)

y0

x

y

Fig. 2


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ff

x1 x2y1 y2

( Domain) ( Range)

y

x0

y1

y2

( x1, y1 )

( x2, y2 )

xx1 x2

Fig. 3a

Fig. 3b

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Since f represents a particular rule of mapping, adifferent functional notation must be employed todenote another function that may appear in thesame model.

The customary symbols (besides f) used for theabove purpose are g, F, G, the Greek letters (phi), (psi), and their capitals, and .

It is permissible to write y = y (x) and z = z (x).

In the function y = f(x), x is referred to as theargument of the function, y is called the value ofthe function. We shall also alternatively refer to xas the independent variable and y as thede endent variable.


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Domain and Range of the FunctionDomain and Range of the Function

The set of all permissible values that x can

take in a given context is known as the domainof the function, which may be a subset of theset of real numbers.

The y value into which an x value is mapped iscalled the image of that x value. The set of allimages is called the range of the function,which is the set of all values that the y variable

will take. The domain pertains to the independent

variable x, and the range has to do with thedependent variable y.

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Example 1: The total cost C of a firm per day is afunction of its daily output Q: C = 150 + 7Q. Thefirm has a capacity limit of 100 units of outputper day. What are the domain and the range ofthe cost function?

Class Assignments:

1.If the domain of the function y = 5 + 3x is theset {x | 1 x 4}, find the range of the functionand express it as a set.

2.For the function y = - x2, if the domain is the ofall nonnegative real numbers, what will itsrange be?


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Polynomial Function:

The constant function is a degenerate

case of what are known as polynomialfunctions.

The word polynomial meansmultiterm, and a polynomial function ofa single variable x has the general form

y = a0 + a1x + a2x2 + +anx

n

in which each term contains a coefficientas well as a nonnegative-integer power ofthe variable x.

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Depending on the value of the integer n(which specifies the highest power of x),we have several subclasses of polynomialfunction:

Case of n = 0: y = a0 [constant function]

Case of n=1: y=a0+a1x [linear function]

Case of n=2: y=a0+a1x+a2x2[quadraticfunction]

Case of n=3: a0+a1x+a2x2+a3x

3 [cubicfunction]

and so forth.


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The superscript indicators of the powersof x are called exponents. The highestpower involved, i.e., the value of n, is

called the degree of the polynomialfunction; a quadratic function, forinstance, is a second-degree polynomial,and a cubic function is a third-degreepolynomial.

When plotted in the coordinate plane, a

linear function will appear as a straightline. When x = 0, the linear functionyields y = a0; thus the ordered pair (0,a0) is on the line.

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The coefficient a1

measures the slope(the steepness of incline) of the line.This means that a unit increase x willresult in an increment in y in theamount of a1.

When a1>0, there is positive slope andan upward-sloping line; if a1


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Linear Functions:

Example 1

Assume that transportation costs for a smalldelivery truck (Y = transportation cost) aredependent on the mileage traveled by the truck(X = mileage).

The specific function is

Y = 54 + 1.29X (General form is Y = a0 + a1X)

Where Y is measured in dollars and X ismeasured in miles.

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OX

X = mileage

54

Y

=transportationcost

Y

Y = 54 + 1.29X


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In this function, a0 = $54, indicating thattransportation costs are $54 whether or not thedelivery truck is in operation. This dollar amount

represents those elements of transportation costnot explained by mileage, such as license fees anda part of depreciation.

The slope of the line, a1, measures the change intransportation cost (in dollars) in response to a 1-mile change in miles traveled.

Since a1

= $1.29, transportation costs increase$1.29 for each additional mile traveled. Similarly,for each 1-mile reduction, transportation costsdecrease by $1.29. The same relationship appliesre ardless of the miles traveled b the truck.

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Example 2

he total cost of manufacturing a product can beexpressed as a linear function of the quantityproduced.

As total costs include fixed costs (such as rent,property taxes, and interest payments) andvariable costs (such as wages, raw materials, etc.),

the following notation can be used for the linearfunction given below:

TC = F + VQ

Where TC = total cost for given number of unitsproduced


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he above function shows a direct relationbetween the quantity produced and total cost,i.e., as quantity increases, total cost increases,

and as quantity decreases, total cost of outputdecreases.

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Class Assignment 1:

A firm assumes that its new shaving cream will besold at the constant retail price of $1.89 per unit.Find the function expressing total revenue from thesale of this new shaving cream. Draw this totalrevenue function in quadrant I of the coordinateaxis.

Class Assignment 2:

A city newspaper has a fixed cost per daily printingof $7000 and a variable cost per copy printed of$0.12. State the total cost function for a dailyprinting of this newspaper. What is the total cost ofa daily printing of 25,000 copies?


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Quadratic Functions:

Quadratic functions have the form

Y = b0 + b1X + b2X2

where b0, b1 and b2 are constants and b2 is notequal to zero.

When graphed on the coordinate axis, aquadratic function is a vertical parabola.

A vertical parabola is characterized by oneturning point where the function changesdirection either from increasing to decreasingor from decreasing to increasing.

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The turning point is called the vertex. Thevertex is either a maximum (Fig. a) orminimum (Fig. b).

Fig. a Fig. bO X O X

YY

If b2 is less than zero, the vertical parabola willappear as in Fig. a (concave downward).


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If b2 is greater than zero, the vertical parabolawill appear as in Fig. b (concave upward).

he X coordinate point of the vertex is foundby setting X equal to -b1/2b2.

Substitution of the resulting X value into thequadratic function yields the value of Y and,therefore, the coordinates of the vertex.

Example 1

he demand for televisions sold by a chain ofretail outlets is expressed by the linearequation where price is a function of thequantity sold.

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P = 720 0.09Q

with P = price in dollars and Q = quantity oftelevisions demanded.

Total revenue (TR) = price (P) . quantity (Q)

P.Q = (720 0.09Q)Q = 720Q 0.09Q2

he above equation is a quadratic total revenuefunction with b0 = 0.

In the expression, b2 is negative, indicating thatthe function is concave downward.

he X coordinate (or quantity consumed) at thevertex is X = -b /2b = (-720)/2(-0.09) = 4000


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X or the quantity of televisions correspondingto maximum total revenue, is equal to

Q = X = 4000 he firm will achieve maximum total revenue

by selling 4000 televisions.

he total revenue corresponding to thisquantity is: TR = 720Q 0.09Q2

720(4000) 0.09(4000)2 = $1, 440, 000

he maximum point of the total revenuefunction specifies a total revenue of $1,440,000and a quantity of 4000 televisions.

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o determine the price corresponding to thisquantity, Q = 4000 is substituted into thedemand function as follows:

P = 720 0.09(4000) = $360

Analysis of the above quadratic function hasshown that the retailer will be able to achieve amaximum total revenue of $1,440,000 byselling 4000 televisions at a price of $360 pertelevision.


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Class Assignment 3:

For the following demand function for shirts

where price (in dollars) is a function of thequantity demanded:

P = 36 0.06Q

a.Find the total revenue function for shirts.

b.Find the vertex of the total revenue function.

c.Show that this point represents maximum totalrevenue.

d.What is the price charged for shirts at thevertex of the total revenue function?

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Class Assignment 4:

Given the following average cost (per unit cost)function for the production of programmablecalculators:

Average cost per unit = 0.06Q2 84Q + 30,000

a.Find the quantity at the vertex of the average-

cost-per-unit function.

b.Find the average cost per unit at the vertex.

c.Is this vertex a maximum point or a minimumpoint? Why?


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Cubic Functions:

It is the polynomial of degree 3 with oneindependent variable.

The form of the cubic function is

Y = b0 + b1X + b2X2 + b3X

3

where b3 is not equal to zero.

A definitive statement about the shape of all cubicfunctions is not possible.

Often, however, a cubic function has two turning

points. Many cubic functions include a segment which is

concave downward and a segment which isconcave upward.

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Example 1

The law of diminishing returns states that whenunits of a variable factor (e.g., labor) are addedto a fixed factor (e.g., land), the outputresulting from additional units of labor firstincreases at an increasing rate, then increasesat a decreasing rate, and finally decreases.

he function used to express this law isreferred to as a production function.

In the present case, production functionspecifies the relation between output and theuse of the variable factor.


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X = Input

Total Production

Y = b0 + b1X + b2X2

+ b3X3

Y

=Outp

ut

Operation of the law of diminishing returns causes thetypical production function to be cubic as there is (1) aconcave upward segment where returns to thevariable factor are increasing at an increasing rate and(2) a concave downward segment where returns tothe variable factor are first increasing at a decreasingrate and then decreasing.

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In the production function, b0

often equals 0 asoutput is not possible without the application ofat least some of the variable input.


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Example 2

he total cost function, expressing total cost as

a function of a firms output, frequently iscubic.

In the present case, there is a concavedownward segment followed by a concaveupward segment.

he total cost of production is related toreturns from the use of variable factors.

In general, if returns from a factor areincreasing at an increasing rate, total costs ofproduction are increasing at a decreasing rate.

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And if factor returns are increasing at adecreasing rate or decreasing, total costs ofproduction are increasing at an increasing rate.

A cubic total cost function has the functionalform Y = b0 + b1X + b2X

2 + b3X3

where Y = total cost

X = quantity of output

b3 0


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X = Output

b0Y

=Totalcost

Total cost:

Y = b0 + b1X + b2X2 + b3X

3

he above graph shows that as output increases,costs increase at a decreasing rate and,

subsequently, increase at an increasing rate asdiminishing returns to the variable factor occur.hus, a concave downward segment follows a

concave upward segment of the curve as outputincreases.

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In the equation for total cost, b0

represents fixedcosts or those costs which do not vary withoutput.

Class Assignment 5:

In the total cost equation, where cost is a functionof the quantity produced C(Q):

C(Q) = -0.2Q3 + 0.03Q2 + 2Q + 1500

1.What cost component is represented by theconstant 1500?

2.Why are there both concave upward andconcave downward segments on a typical totalcost curve?


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Rational Functions:

A function such as

y = (x -1) / (x2 + 2x + 4)

In which y is expressed as a ratio of twopolynomials in the variable x, is knownas a rational function. According to thisdefinition, any polynomial function mustitself be a rational function, because it

can always be expressed as a ratio to 1,which is a constant function.

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A special rational function that hasinteresting applications in business/economics is the function

y = a/x or xy = a

Which plots as a rectangular hyperbola.

This function may be used to represent

that special demand curve with price Pand quantity Q on the two axes forwhich the total expenditure PQ isconstant at all levels of price. Such ademand curve is the one with a unitaryelasticity at each point on the curve.


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Another application is to the averagefixed cost (AFC) curve. With AFC onone axis and output on the other,the AFC must be rectangular-hyperbolic because AFC X Q (= totalfixed cost) is a fixed constant.

The rectangular hyperbola drawnfrom xy = a never meets the axes.

The curve approaches the axesasymptotically. The axes constitutethe asymptotes of this function.

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y

ox

Slope = a1

y= a0+a1 x

Linear

a0

0 x

y

a0

y= a0+a1 x +a2x2

( Case of a2 0 )

Quadratic


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o

x

y

a0

y= a0+a1 x +a2x2 + a3x

3

Cubic

y

xo

(a>0)

Rectangular-hyperbolic

y = a / x

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Exponential

y = bx

0 x

y

(b> 1)

Logarithmic

y = logb x

0 x

y


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Algebraic Functions:

Any function expressed in terms of

polynomials and/or roots (such as squareroot) of polynomials is an algebraicfunction. Accordingly, the functionsdiscussed far are all algebraic. A functionsuch as y = x2+3 is not rational, yet it isalgebraic.

Nonalgebraic Functions:

Exponential functions such as y = bx, inwhich the independent variable appearsin the exponent, are nonalgebraic.

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(0, 1)

y = 6xy = 4x y = 2x

Fig. Exponential function where the value of b is greater than one.


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In the above figure, for each of the functions,the value of the function y approaches the Xaxis as x approaches -. This indicates y

approaches 0, but never equals it, as xdecreases.

The exponential function y = bx when b isgreater than 1 is asymptotic to the X axis as xapproaches -.

y is continually increasing for increasing values

of x. So, y = bx when b is greater than 1 is amonotonically increasing function of x.

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For example, automobile sales may risecontinually as national employment increases.So, automobile sales are a monotonicallyincreasing function of national employment.


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Just as the function y = f(x) maps a point inthe domain into a point in the range, thefunction g will do precisely the same. The

domain is in this case a set of ordered pairs (x,y), because we can determine z only whenboth x and y are specified.

he function g is a mapping from a point in atwo-dimensional space into a point on a linesegment (i. e., a point in a one-dimensional

space), such as from the point (x1, y1) into thepoint z1 or from (x2, y2) into z2.

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he domain of the function will be some subsetof the points in the xy plane, and the value ofthe function (value of z) for a given point in thedomain say (x1, y1) can be indicated by theheight of a vertical line planted on that point.

he association between the three variables issummarized by the ordered triple (x1, y1, z1),

which is a specific point in the three-dimensional space.

The function y = f(x) is a set of ordered pairs,the function z = g(x, y) will be a set of orderedtriples.


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0

z

x1

x2

y2

y1

(x1, y1, z1)

(x2, y2, z2)

x

y

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The ordered quadruple (u1, v1, w1, y1) is apoint in the four-dimensional space. The locusof such points will give the (nongraphable)graph of the function y = h (u, v, w), which ishypersurface.

These terms, viz., point and hypersurface, arecarried over to the general case of the n-dimensional space.

Functions of more than one variable can beclassified into various types. For instance, afunction of the form

y = a1x1 + a2x2 ++ anxn is a linear function.

biz - quatitative.managment.method chapter.04

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