Functions: Concept of function, Domain and Range of function, Types of function, Business application of function, Illustrative examples. A.P. and G.P.: Difference between A.P. and G.P., Definition of A.P. series, Nth term of A.P., Sum of n terms of A.P., Definition of G.P., Nth term in G.P., Sum of n terms of G.P., Illustrative examples. Matrix: Definition, Order of Matrix, Types of Matrix, Matrix operations, Special types of matrix, Matrix on the basis of transpose, Determinants: order of determinants, Minor and cofactors, Properties of determinants, Adjoint of square matrix, Inverse of square matrix, Simultaneous linear equation: Cramer’s Rule Inverse Method, Economic application of matrix. Illustrative examples.

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Quantitative Techniques
Volume-1
(Revised)
Functions, (A.P., G.P.), and Matrix
E-Book Code : QTVOL1
by
Narender Sharma
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Quantitative Techniques
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Click on Contents
Functions ............................................................................................................................................................................................................. 5
Concept of Functions .................................................................................................................................................................................. 5
Domain and Range of Function .............................................................................................................................................................. 5
Types of Functions ...................................................................................................................................................................................... 5
Explicit and Implicit Functions : ....................................................................................................................................................... 5
Single – valued and multi-valued Functions: ............................................................................................................................. 5
Increasing and Decreasing Functions: ........................................................................................................................................... 6
Even and Odd functions : ..................................................................................................................................................................... 6
Polynomial Functions : ......................................................................................................................................................................... 6
Rational Function: .................................................................................................................................................................................. 6
Composite Function:.............................................................................................................................................................................. 6
Exponential Function: ........................................................................................................................................................................... 6
Logarithmic Function: .......................................................................................................................................................................... 7
Constant Function: ................................................................................................................................................................................. 7
Identity Function .................................................................................................................................................................................... 7
Absolute value Function: ..................................................................................................................................................................... 7
Signum Function: .................................................................................................................................................................................... 8
Greatest Integer Function: .................................................................................................................................................................. 8
Reciprocal Function: .............................................................................................................................................................................. 8
Business Application of Functions/Functions in Economics .................................................................................................... 9
Demand Function: .................................................................................................................................................................................. 9
Supply Function: ..................................................................................................................................................................................... 9
Total cost function : ............................................................................................................................................................................ 10
Production Function: ......................................................................................................................................................................... 10
1. Cobb-Douglas Production Function: .................................................................................................................................. 10
2. CES Function : .............................................................................................................................................................................. 10
Revenue Function: ............................................................................................................................................................................... 11
Profit Function : ................................................................................................................................................................................... 11
Consumption Function: ..................................................................................................................................................................... 11
Arithmetic Progression(A.P.) And Geometric Progression(G.P.)........................................................................................... 13
Difference between A.P. and G.P. ...................................................................................................................................................... 13
Greedy Merchant: A story about Magic of G.P. ............................................................................................................................. 13
Arithmetic progression .......................................................................................................................................................................... 14
Nth term of A.P. ..................................................................................................................................................................................... 14
Sum of n terms of A.P. ........................................................................................................................................................................ 15
Geometric Progression ........................................................................................................................................................................... 18
Nth term of G.P. ..................................................................................................................................................................................... 18
Sum of n terms in G.P. ........................................................................................................................................................................ 19

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Matrix ................................................................................................................................................................................................................. 22
Definition: ............................................................................................................................................................................................... 22
Order of matrix : ................................................................................................................................................................................... 22
Types of Matrix .......................................................................................................................................................................................... 22
Rectangular Matrix: ............................................................................................................................................................................ 22
Square Matrix: ....................................................................................................................................................................................... 22
Diagonal Matrix: ................................................................................................................................................................................... 22
Scalar Matrix: ......................................................................................................................................................................................... 22
Identity(Unit) Matrix : ....................................................................................................................................................................... 22
Null Matrix or Zero Matrix : ............................................................................................................................................................ 22
Row Matrix: ............................................................................................................................................................................................ 22
Column Matrix : .................................................................................................................................................................................... 22
Upper Triangular Matrix : ................................................................................................................................................................ 22
Lower Triangular Matrix : ................................................................................................................................................................ 23
Sub Matrix: ............................................................................................................................................................................................. 23
Trace of a Matrix : ................................................................................................................................................................................ 23
Equality of Two Matrices : .............................................................................................................................................................. 23
Matrix Operations ................................................................................................................................................................................... 23
Addition or Subtraction of Matrices : .......................................................................................................................................... 23
Scalar Multiplication : ........................................................................................................................................................................ 23
Properties of Matrix Addition ......................................................................................................................................................... 23
Matrix Multiplication.......................................................................................................................................................................... 24
Properties of Matrix Multiplication .............................................................................................................................................. 24
Special Types of Matrices ...................................................................................................................................................................... 24
Idempotent Matrix : ........................................................................................................................................................................... 24
Nilpotent Matrix : ................................................................................................................................................................................. 24
Involutory Matrix : .............................................................................................................................................................................. 24
Transpose of Matrix : ......................................................................................................................................................................... 24
Matrix on the basis of Transpose ....................................................................................................................................................... 25
Symmetric Matrix : .............................................................................................................................................................................. 25
Skew -Symmetric Matrix : ................................................................................................................................................................ 25
Orthogonal Matrix: .............................................................................................................................................................................. 25
Determinants : ................................................................................................................................................................................................ 25
Determinant of order one : .............................................................................................................................................................. 25
Determinant of order two: ............................................................................................................................................................... 25
Determinant of order Three:........................................................................................................................................................... 25
Minors and Cofactors: ...................................................................................................................................................................... 25
Properties of Determinant ............................................................................................................................................................... 26
Adjoint of a Square Matrix .................................................................................................................................................................. 26

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Inverse of a square Matrix : ................................................................................................................................................................. 27
Simultaneous Linear Equations : ....................................................................................................................................................... 28
Cramer’s Rule ........................................................................................................................................................................................ 28
Inverse method ..................................................................................................................................................................................... 32
Economic Applications of Matrices ................................................................................................................................................... 34
References ................................................................................................................................................................................................... 36
Feedback ...................................................................................................................................................................................................... 36

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Functions
Concept of Functions
The term ‘functions’ refers to the relationship between the variables. When change in one variable is observed due to change in the other variable , then it is said to have functional relationship. In other words , a variable `y’ is said to be a function of another variable `x’ when to every value of `x’ there corresponds a definite value of `y’. It is written as : ( )
Here `x’ is called independent variable and `y’ is called dependent variable.
Domain and Range of Function
The set of values which an independent variable can assume is called the domain of the function and the set of corresponding values of y(i.e. dependent variable) is called the range of the function.
Functions are helpful to understand the relation among different Input and Output variables of Business.
Types of Functions
Functions are classified into the following categories:
Explicit and Implicit Functions :
A function y f(x) is said to be an explicit function of x if the value of y is directly given in terms of x. It is written as y f(x)
But if the variables y and x are mixed in the functional relationship, then the function is said to be an implicit function It is written as : f(x,y)
For example, y x is an explicit function,
wheres x y a is an implicit function.
Single – valued and multi-valued Functions:
A function ( ) is said to be single valued function of x if to every value of x, there corresponds one and only one value of y. e.g. , log are all single valued function of x. And a function ( ) is said to be many valued function of x if a value of x gives rise to more than one value of y. e.g. is an example of many valued function.
Key Process Input Variables(KPIV)
( )
Key Process Output Variables(KPOV)
 Independent
 Input
 Cause
 Problem
 Control
 Factor
 Dependent
 Output
 Effect
 Symptoms
 Monitor
 Response

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Increasing and Decreasing Functions:
A function ( ) is said to be an increasing function of x if the value of y always increases as x increases. The function ( ) is said to be a decreasing function of x if the value of y always decreases as the value of x increases. Normal demand functions and total cost functions are decreasing and increasing functions respectively.
Even and Odd functions :
A function ( ) is said to be an even function of x if ( )= ( ). e.g. ( ) = is even function. Similarly a function ( ) is said to be an odd function of x if ( )= ( ).e.g. ( ) = is an odd function.
Polynomial Functions :
A function of the form
( ) …………………….…
where n is a positive integer and , is called a polynomial function of degree n.
If n=0, we have ( ) , a constant function
If n=1 then ( ) , a linear function
If n=2 then ( ) , a quadratic or parabolic function.
If n=3 then ( ) , a cubic function and so on .
If …………….. ,
then ( ) (take ), a power function.
Rational Function:
A rational function is defined as the ratio of two polynomials. e.g. ( ) ( ) is a rational function where ( ) ( ) are polynomials in x and ( ) .
e.g. f(x) x x x x is a rational function.
Composite Function:
If y is a function of u and u is a function of x , then y is said to be composite function of x . e.g. if y f(u)and u g(x),then y f*g(x)+ is a composite function of x
Exponential Function:
Exponential function is a function of the form . When a >1 , the exponential function is increasing and when 0<a<1, the exponential function is decreasing.
when a>1
0
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y x

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when 0<a<1
Logarithmic Function:
Logarithmic function are inverse to exponential functions. The function ( ) log is called logarithmic function.
Constant Function:
A function of the type ( ) where is a fixed real number e. g. ,
Identity Function
( ) , is called the identity function. As represents a straight line passing through the origin and inclined at an angle with x axis. To draw a line, it is sufficient to have any two pts on the line, the graph of can also be drawn by two different values of x.
Absolute value Function:
( ) ( )
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0
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f(x)
x
Log x