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CHAPTER

2

Functions and Their

Applications

Chapter Outline

IntroductionSome Useful Functions in Business and

The Concept of a FunctionEconomics

Types of FunctionsEquilibrium of an Economic System

Roots (Zeros) of a FunctionBreak-Even Analysis

Learning Objectives

After reading this chapter, you should be able to

Define

types of functionsroots (zeros) of a function useful functions in business and economicsequillibrium of an economic system

Explain

concept of a function the relationship between roots and coefficients of a quadratic equation break-even point and its explanation.

INTRODUCTION

For the use of mathematical models in decision-making the first requirement is to identify relevant factors

(also called variables) involved in the problem and then defining their interrelationships. S uch relationships are expressed in the form of an equation or set of equations/inequalitie s. These equations or inequalities

with or without an objective function help the decision-maker in better understanding of the problem and

arriving at an optimal decision. For example, total inventory cost is expressed in terms of total purchase

cost, ordering cost, holding cost and shortage cost. The differential calculus method is used to calculate

economic order quantity to achieve minimum total inventory cost. The aim of this chapter is to explain some fundamental concepts about fu nctions, their classification and application in the context of business and economic problems.

55Functions and Their Applications

have a specific relationship among the selected variables. For example, for the purpose of finding total

inventory incremental cost (TIC), the specific relationship between T and Q is stated as : TIC = ph ...(i) where Q ph p and C h are total annual demand, procurement

cost per order and holding cost per unit per time period respectively. It may be noted here that equation

(i) indicates rule of correspondence between the dependent variable TIC and independent variable (Q). That

is, as soon as different values to Q are assigned in the set of real num bers, the corresponding unique value

of TIC is determined by the given relationship and that relationship is called a real function. The various

values o form a set called the domain and the corresponding values of TIC form another set called the

range of the function. It can also be expressed as f : Q TIC. Based on the above discussion, we can now define the function as a corre spondence among variables of two non-empty sets A and B as follows : Definition : If A and B are two non-empty sets and there exists a rule of corresponde nce by which each

element x of set A is related to unique element y of set B, then such correspondence is called the function

from A to B. It is represented as : f: A B, such that y B and x A. wherey=f (x)

Remarks

1.Value of function : The element y in B that is associated to x by f is denoted by f (x) and is called

the value of f at x.

2.Domain o : The set A is called the domain of function f.

3.Co-domain of f : The set B is called the co-domain of function f.

4.Range o : The set {f (x) : x A, f (x) B} of all values taken by f is called the range of f. It is

obviously a subset of B.

Intervals

If a and b are two real numbers such that a > b , then a set of real numbers can be enumerated between

a and b.

The set of all real numbers between a and b without these end points is called the open interval and

is written as : (a, b)={x R : a < x < b}

However, if end points a and b are included in the set, then it is called a closed interval and is written as:

[a , b]={x R : a x b] There are also intervals which are closed at only end point. For example, (a, b]={x R : a < x b} and[a, b)={x R : a x < b} Every mathematical relationship may not define the function. For example , the equation y = x does not

define a function, since we find that there exist twovalues ± 2 of y corresponding to the given value x = 4.

56Business Mathematics

The dimension of a function is determined by the number of independent variables. For example, (a)y = f (x) is a single-variable (or one-dimensional) function. (b)y = f (x, z) is a two-variable (or two-dimensional) function. (c)y = f (x, z, r) is a three-variable (or three-dimensional) function.

Example 1 : Let p + 3q/2 = 27 be an equation involving two variables p (price) and q (quantity). Indicate

the meaningful domain and range of this function when (a) the price (b) the quality are considered independent variables. Solution : (a) When price (p) is taken as independent variable, we have q= 18 - p

Domain :0 p 27

Range :0 q 18

(b) When quantity (q) is taken as independent variable, we have p= 27 - q

Domain :0 q 18

Range :0 p 27

Example 2 : Find the domain and range of the following functions : (a) y = - |x| (b) y = x. Solution : (a) Domain= set of values of x for which y is meaningful = set of real numbers (R) because for all x R, y has a unique value, i.e., |x| 0 for all x R.

Thus, -

x0 or y 0, for all x R

Range : {

y : y= - x and x R} = {y : y 0}

(b) The value of y will be real for those values of x for which the expression under the radical sign is

non-negative ( 0). Thus, the domain of y is given by 3 - 2x 0 or 3 2x or x 3/2, i.e., the set of all real numbers in the interval (, 3/2)]. For range, we have y = x or y 2 = 3 - 2x which gives x = (3 - y 2 )/2.

Since x 3/2, therefore, (3 - y

2 )/2 (3/2) or 3/2 - y 2 /2 3/2 or - y 2 /2 0 or y 2 /2 0 which implies

either y 0 or y 0. Also y is the positive square root, therefore rejecting y 0, we have y 0. Thus,

the range of y is the set of all non-negative real numbers. Example 3 : Find the domain and range of the following: (a) x x (b)[x](c)[x] - x(d)|x - 1 |

Solution. (a) For f (x) =

xx , where f is defined for all x except at x = 0. Therefore, domain (f) = R - {0} when x > 0, f (x) = 1 and when x < 0, f (x) = - 1. Thus, range (f) = {1, - 1}.

59Functions and Their Applications

|x| = xx xx x 0 0 00

Step-function : If a function is defined on a

closed interval [a, b] and assume a constant value in the interior of each sub-interval say [a, x 1 ], [x 2 , x 3 ],..., [x n , b] of [a, b] where a < x 1 < x 2 x n b , then such function is called a step function. Symbolically, it may be ex- pressed as : y or f (x) = k i for all values of x in the i th sub-interval. The graph of this function is given in Fig. 2.3, y 1 < y 2 < y 3 Convex Setand ConvexFunc tion: A set S of points in the two-dimensional plane is said to be convex if for any two points (x 1 , y 1 ) and (x 2 , y 2 ) in the set the line segment joining these points is also in the set.

Mathematically, this definition implies (x

1 , y 1 ) and (x 2 , y 2 ) are two different points in S, Then the point whose coordinates are given by x 1 + (1 - ) x 2 ; y 1 + ( - ) y 2 } ; 0 1 must also be in the set S.

If = 0, then we get the coordinates (x

2 , y 2 ) of the given point. But, if = 1/2, the corresponding point on the line segment is x xyy 1 212 22
x 1 , y 1 ) and ( x 2 , y 2

The typical examples of a convex set are a circle and triangle. Figure 2.4 (a) and (b) illustrates the

example of convex and non-convex sets. a) Convex Set(b) Non-Convex Set

Fig. 2.4 :

Convex and non-Convex Set

Fig. 2.3 : Step Function

Fig. 2.2 :

Absolute Value of a Function

60Business Mathematics

A function f(x) defined over a convex set S

is said to be convex function if for any two distinct points x 1 and x 2 lying in S for any 0 1, f{ x 1 + (1 - )x 2 } f(x 1 ) + (1 - ) f(x 2

The graph of this function is given in

Fig. 2.5.

Inverse Function : If variables x and y are inter- dependent such that (i) y is the function of x; y = f(x) (ii) If x is the function of y, x = g(x), then f is known as the inverse of g and vice versa.

For example, if y = x

2 + 2x + 6, then the inverse function is: x = - 1 ± y6 y=f(x)=f[g(y)] orfog(y) andx=g(y)=g[f(x)] orgof (x) Consequently, fog = gof. Functions fog and gof are known as composite functions.

Rational Function :

A rational function is defined as the quotient of two polynomial functio ns and is of the form : y= ax ax a bx bx b nnnn m mmm 110
1 10 x x , Q ( x ) 0

where P (x) is a polynomial function of degree n and Q (x) is a non-zero polynomial function of degree m.

The function

y is defined for all values of x provided the denominator does not become zero. For example, y= x xx is a rational function.

An expression which involves root extraction on terms involving x is called an irrational function. The

functions such as x, xx are examples of irrational function.

Algebraic Function : A function consisting of a finite number of terms involving powers and roots of the

variable x and the four basic mathematical operations (addition, subtraction, multiplication and division) is

called an algebraic function. In general, it can be expressed as y n + A 1 y n- 1 + ... + A n = 0 where A 1 , A 2 , ..., A n are rational functions of x.

There are two categories of algebraic functions namely : explicit and implicit algebraic functions. For

example, y = x + 3x 3 is an explicit algebraic function, whereas xy 2 + xy + x 2 = 0 is an implicit function. Transcendental Function: All functions which are not algebraic are called transcendental function s. These functions include

1.Trigonometric Functions. The trigonometric functions of an angle ( be any real number) are given by:

sin = sin c , cos = cos c , tan = tan c cosec = cosecquotesdbs_dbs14.pdfusesText_20

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