Fundamentals of Mathematics

Fundamentals of Mathematics for JEE Main and Advenced
Functions and Graphs
Sanjay Mishra
B. Tech.
Indian Institute of Technology
Varanasi

CONTENTS

*********** Chapitre I *********
Relations 1.1-1.56
Introduction 1.1
Cartesian Product of Two Sets
Number of Elements in Cartesian
ProductAxB *Properties and Laws of Cartesian Product
Basic Definitions 1.7
Relations 1.8
Domain, Co-domain and Range of Relation 1.9
Universal Relation from set A to set 8 1.10
Number of Relations from setA to set B 1.10
Relation on a Set 1.10
Representation of Relation in Different Forms 1.10
Classification of Relations 1.13
*One-One or Injective Relation
Into Relation 1.14
* One-One-Onto Relation (Bijective Relation)
Types of Relations 1.20
*Reflexive Relation *Identity Relation *Transitive Relation
*Anti-symmetric Relation *Equivalence Relation
Composition of Relations 1.27
Inverse of a Relation 1.27
Important Remarks at a Glance 1.28
Multiple—Choice Questions 1.32
Tutorial Exercise 1.39
Answer Keys 1.43
Hints and Solutions 1.44

*********** Chapitre II *********
Functions 2.1—2.376
Introduction 2.1
Definition of Function 2.1
*Pictorially
Domain, Co-Domain and Range of Function 2.3
Representation of Functions 2. 13
Function as a Set of Ordered Pairs 2.15
Parametric Representation 2.16
Diagrammatic Representation 2.19
Mathematical Tools Used to Find the Domain and Range of Functions 2.29
*Laws of Inequality 0 Conclusion 0 Conclusion
Polynomial Expression 2.33
Solving Rational Inequalities 2.33
Wavy Curve Method 2.33
Increasing and Decreasing Function 2.38
Monotonic Function 2.39
Interval of Monotonicity 2.39
To Find Interval of Monotonicity for y =j(x) 2.39
Strictly Monotonic Functions 2.40
Some Standard Functions and their Properties 2.46
*Algebraic Function
Polynomial Function 2.46
Rational Function 2.49
Irrational Function 2.50
Power F unetion 2.50
Transcendental Function 2.54
Modulus Function 2.54
Signum Function 2.62
Constant Function 2.63
Identity Function 2.64
Equal or Identical Functions 2.65
Exponential F unetion 2.68
Properties of Exponential Function 2.68
Solving Exponential Equations 2.69
Solving Exponential Inequality 269
Composite Exponential Function 2.70
Hyperbolic Functions 2.70
Logarithmic Function 2.73
Properties of Logarithmic Functions 2.74
Greatest Integer Function (Bracket Function) 2.84
Properties of Greatest Integer Function (Bracket Function) 2.84
Least Integer Function 2.95
*Properties of Least Integer Function
Fractional Part Function 2.95
*Properties of Fractional Part Function
Nearest Integer Function 2.100
*Properties of Neares Integer Function
Method of Solving Inequality Involving x, {x} and [x] 2.101
Binary Operation 2.105
*Definition of Binary Operations
Properties of Binary Operation * On a Set A 2.109
Domain of Functions 2.116
Properties of Domain 2.116
Range of Functions 2.122
*Bounded and Uri-bounded Functions *Greatest Lower Bound *Conclusion
*Least Upper Bound *Conclusion *Intermediate Ihlue Theorem
Definition of Range 2.124
*Methods to Find Range of Functions
Classification of Functions 2.140
One-One/Many-One Functions 2.140
*One-One (Injective) Function
Many-One Functions 2.140
Method of testing for Injectivity 2.140
Suijective and Non-Surjective Function 2.150
*Onto (Surjective) Function
Into (Non-Surjective) Function 2.150
Hit and Trial Method 2-151
One-One Onto Function (Bijective Function) 2.154
Number of Relations and Functions 2.157
Composition of Functions 2.163
*Composite of Uniformly Defined Functions
Properties of Composition of Functions 2.168
Composition of Non-Uniformly Defined Functions 2.178
Invertible/Non-Invertible functions 2.185
*Definition of Inverse ofa Function
Conditions of Invertibility of a Function 2.186
Conclusion 2-186
* IUethod to Find Inverse of a Given Function
Properties of Inverse of a Function 2.191
Even and Odd Functions 2.202
One-One/Many-One Functions 2.140
*One-One (Injective) Function
Even Functions 2.203
Odd Functions 2.205
Properties of Odd Functions 2.206
Algebra of Even-Odd Functions 2.209
Even and Odd Extension of Function 2.213
*Even Extension *Odd Extension
Periodic Functions 2.220
*Definition of Periodic Function *Test of Periodicity
Some Important Facts 2.222
Properties of Periodic Function 2.227
Period of Composite Functions 2.228
Periodicity of Modulus/Power of a Function 2.230
Periodicity of Product/Division of Function 2.236
Exception to LCM Rule 2.237
Periodicity of Functions Expressed by Functional Equations 2-241
Multiple-Choice Questions 2.247
Tutorial Exercise 2282
Answer Keys 2.94
Hints and Solutions 2.296

*********** Chapitre III *********
Graph Theory
Introduction 3.1
Graphs of Function 3.1
Algebraic Functions 3.1
*Polynomial Function
Rational Functions 3.4
Irrational Function 3.5
*Graph of Some Basic Irrational Function
Piece-Wise Defined Functions 3.7
Transcendental Functions 3.9
*Trigonometric Curves
Exponential Function 3.11
Logarithmic Function 3.12
Inverse Circular Functions 3.13
Standard Conic Sections 3-17
*Circle
Representation as a Function 3.17
*Parabola 0 Ellipse 0 Hyperbola
Transformation of Graphs 3.29
*Transformation of Graph Continued 0 Graph of y = {f{x}}
Inequality 3.78
*Linear Equation and Inequality
Algorithm 3.79
Graphs of Reciprocal Function 3.89
Addition of Graphs 3.90
Multiplication of Graphs 3.95
Special Case of Enveloped Graphs 3.95
Maximum and Minimum Functions 3.98
Sketching of Non-Standard Curves 3.107
SymmetnrlMonotonicity/Curvature of Function 3.107
Asymptotes 3.118
*Definition
Horizontal Asymptote (Asymptotes Parallel to x-axis) 3.120
Procedure to Find Horizontal Asymptote to Algebraic Curve 3.120
*Algorithm
Vertical Asymptotes (Asymptotes Parallel to y-axis) 3.121
Procedure to Find Vertical Asymptote 3.123
Oblique Asymptotes 3. 123
*Procedure to Find Oblique Asymptotes
Another Method to Find Oblique Asymptote for Second Degree Curve 3.124
Method to Find Oblique Asymptotes for Algebraic Curves of any Degree 3.125
Asymptote by Expansion 3.126
The Position of the Curve with Respect to an Asymptote 3.127
Singular Points (Multiple Points) 3.128
*Introduction 0 Definition
Double Points 3.128
*Types of Double points
Necessary Conditions for the Existence of Double Points 3.129
Types of Cusps 3.129
Multiple-Choice Questions 3.140
Tutorial Exercise 3.160
Answer Keys 3.170
Hints and Solutions 3.171