ABN Functions

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NAMASTE gd:]t Welcome To Nepal Mathematics Centre Lecture Series in Lessons in Basic Mathematics Nepal Mathematics Centre HjHjnkf GOOD MORNING Namaste ( National Mathematical Sciences Team ) …

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NMC NEPAL MATHEMATICS CENTRE Nepal Mathematics Centre is a non-profit service oriented educational foundation/Trust dedicated to the improvement of teaching, learning, evaluation, research and applications of mathematics in Nepal. Its short form is NMC. In Vernacular , it will be called “ g]kfn ul0ft s]Gb|” -g]us]_ . NMC NEPAL MATHEMATICS CENTRE g]kfn ul0ft s]Gb| 2009 @)^^ D.A.O.K. Regd. No. 146/2067

NMC's main objectives are: 

To launch a nationwide Mathematics Awareness Movement in order to convince the public in recognizing the need for better mathematics education for all children, To initiate a campaign for the recruitment, preparation, training and retaining teachers with strong background in mathematics, NMC's main objectives are

NMC's main objectives (Continued): 

To help promote the development and dissemination of innovative ideas, methods and materials in the teaching, learning and research in mathematics and mathematics education, To provide a forum for free discussion on all aspects of mathematics education, To facilitate the development of consensus among diverse groups with respect to possible changes, and To work for the implementation of such changes. NMC's main objectives (Continued)

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NEPAL MATHEMATICS CENTRE Presents Relations Functions Graphs

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Moderators Anjana Pokharel Bindra Sakya Neelam Subedi

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Unit Two Unit Title Relations Functions Graphs Sub-Units 2.1 Relations 2.2 Functions 2.3 Graphs

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Sub-Unit FUNCTIONS

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Learning Objectives To narrate briefly the origin and development of the notion of function To know what a function, domain and range are, To find the domain and range of a function, To identify if a relation is a function or not, To represent function in various standard forms

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Learning Objectives To evaluate functional values, To apply vertical line test, To define standard types of functions ( Injective, surjective, bijective, extension and restriction) To determine the type of function.

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Learning Objectives To identify the difference between one-one (injective), many-one (surjective), and one-one and onto (bijective) functions . To know what inverse image of a point and inverse function To compute inverse image and inverse of a function To know how to find the composition of a function.

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Learning Objectives To define real-valued function of a real variable To apply the fundamental operations on real-valued functions To define simple algebraic functions (linear function, quadratic function, cubic function) To describe the basic characteristics of the graphs of the algebraic functions

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Learning Objectives To define elementary transcendental functions (Trigonometric functions, exponential function, logarithmic function) To describe the basic characteristics of the graphs of the elementary transcendental functions

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To use the exponent and e keys on your calculator. To evaluate an exponential function. To graph exponential functions. To calculate compound interest problems Learning Objectives

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To know the definition of a logarithmic function. To write a log function as an exponential function and vice versa. To graph a log function. To evaluate a log. To find the domain of a log function. Learning Objectives

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Methods of Instruction/Instructional Strategies: Lecture-Discussion Internet,

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Suggested Strategies for Assessment: Exams/tests/quizzes Written assignments

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Additional Learning Resources 1. D.B .Adhikary, N.Kshetri, Elements of Mathematics, Part I, Himalayan Book Stall (1999), Kathmandu, Nepal D.R. Bajracharya, An Introduction to Basic R.M.Shreshtha,… Mathematics, Vol. 1, Sukunda Pustak Bhavan (2065/2008), Kathmandu, Nepal. D.R. Bajracharya, Higher Secondary Level Basic R.M.Shreshtha,… Mathematics, Grade XI, Sukunda Pustak Bhavan (2065/2008), Kathmandu, Nepal. 4. G.D. Pant Certificate Mathematics, Part 1, Nepal Sahitya Prakashan Kendra (2054), Kathmandu, Nepal. 5. K.N. Pandey, Foundation of Mathematics, Class XI, K.P.Ghimire Pigeon Books (2064), Kathmandu Nepal. 6. P.M. Bajracharya, Fundamentals of Mathematics, Grade XI, G.B.Basnet Budha Academic Publishers and Distributors (2064), Kathmandu, Nepal.

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Additional Learning Resources 7. P.M.Bajracharya Real Analysis, Budha Academic Publishers and Distributors(2064), Kathmandu, Nepal. 8. R.M. Shreshtha Fundamentals of Mathematical Analysis, Sukunda Pustak Bhavan (2065/2008), Kathmandu, Nepal. 9. S.M.Maskey Principles of Real Analysis, Bhudipuran Prakashan (2061), Kathmandu, Nepal. 10. S.M.Maskey Introduction to Modern Mathematics, Vol. 1 Padma Educational Traders, Kathmandu, Nepal S. R.Pant, A Text Book of Higher Secondary P.R. Adhikary,… Mathematics (2004), Budha Academic Publishers and Distributors (2064), Kathmandu, Nepal 12. S.P.Koirala … Higher Secondary Mathematics (1966), Kathmandu, Nepal.

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Additional Learning Resources Namaste - Power Point Presentations - NMCMini-Math Series - NMC Lecture Notes 2. Author - 3

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2.2 Functions Description This sub-unit is designed as an introduction to Definition, Domain and range of a function, Functions defined as mappings, Inverse function, Composite function, functions of special type (Identity, Constant, Absolute value, Greatest integer), Algebraic (Linear, quadratic and cubic), Trigonometric, Exponential logarithmic functions Sub-unit Two

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Recommended Text Namaste : Lessons in Basic Mathematics XI (Paluwa Publication) Pre-Requisite Knowledge of School Mathematics Sub-unit Two

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What is the meaning of function in math ? A. I really don't know the meaning ! -Yahoo : Ask – Answer - Discover A Bit of History

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Function idea originated when early man started to put aside a pebble for every passing sheep and a couple began to rear their child . One-to-0ne Relation Many-to-0ne Relation

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12/15/2011 25 The Persian mathematician , Sharaf al-Dīn al-Tūsī , is reported to have used the idea of a function in the 12 th century As a mathematical term, "function" was coined by Gottfried Leibniz in 1694, latter formalized by Leonhard Euler 1707-1783). He introduced the notation y = f ( x ) for a function

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12/15/2011 26 Lobachevsky is traditionally credited with independently giving the modern "formal" definition “ A function as a relation in which every first element has a unique second element. ” The Bourbaki group named the three types of functions: ( one-one function, onto function , one-one and onto functions ) as injective function, surjective function and bijective function respectively.

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Some Animated Illustrations

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Input ( Fruit) Process ( Squeezing) Output ( Fruit juice)

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Input (Fruit ) Function (Squeezing) (Squeezing) Output (Fruit Juice ) Apple Banana Cherry . . Grape Apple Juice Banana Juice Cherry Juice . . Grape Juice Input (Fruit ) Output (Fruit Juice ) Function (Squeezing)

A Bit More Mathematical: 

A Bit More Mathematical Rule in Words (Function) Add 7 to the Value x Arrow Mappin g/ Diagram Table Graph Equation : y = x + 7 Input x Output x + 7

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Path of a Bouncing Ball 16 8 4 2 1 0 Decreasing Height ( x ) Difference | x - 0 | smaller and smaller approaching 0 “tends to 0” In symbol x  0 0

Domain: 

Domain Permissible x value a b ● ● Domain [ a , b ]

Function Domain: 

Function Domain ( x , f ( x )) x f ( x ) x-axis y-axis O Domain of f f ● ●

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R a n g e Range Possible y value Range [  ,  ] ● ●  

Function Range: 

Function Range ( x , f ( x )) x f ( x ) x-axis y-axis O f Ra n g e ● ●   [  ,  ]

Function Domain & Range: 

Function Domain & Range ( x , f ( x )) x f ( x ) x-axis y-axis O Domain of f f R a n g e ● ● ● ● [ a , b ] [  ,  ]

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R a n g e R a n g e x-axis y-axis 0 a b x  Domain [ a , b ] Function ( f ) Function   f : [ a, b ]  [  ,  ]. y = f ( x )  [  ,  ]

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R a n g e R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) Piece-wise Defined Function   f = f ' + f '' : [ a, c )  [ c, b ] where f ' : [ a, c )  [  ,  ] and f '' : [ c, b ]  [  ,  ] c

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Formal Definition A function is a rule or correspondence which associates to each element x in a set A a unique element y in a set B . As in the past, we may call x an independent variable , and y the dependent variable.

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Well known notations for functions are as follows: y = f ( x ), x  A , y  B, f : A → B , or, x  f ( x ). The terms mapping, transformation and operator are also used to denote functions . Function Notations

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By domain of the function, f : A → B , we mean the set of all values of x  A; and the set of corresponding values of y  B or images of x  A, is called the range of the function. Domain and Range Sometimes the term codomain or image set is also used to mean range .

Vertical Line Test: 

Vertical Line Test A graph in a coordinate plane represents a function if and only if it intersects the graph at only one point .     x-axis y-axis Graph O

Three points of intersection: 

    Each of the first three vertical lines cuts the following graph at one and only one point, but the fourth one cuts it at three points. So the graph does not represent a function   Three points of intersection x-axis y-axis O

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In practice: A function is usually denoted by letters such as f , g ,  , etc, the corresponding function values or images of x  A by f ( x ), g ( x ),  ( x ), etc. Here, f ( x ), g ( x ), f ( x ), g ( x ),  ( x ) … etc. are read ‘ f of x ’, ‘ g of x ’, ‘ of x ’, etc. Function and Function Values

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R a n g e R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) Function Value At Left End-Point f ( a )   f : [ a, b ]  [  ,  ]. a  [ a, b ]  f ( a ) is defined at left end-point [  ,  ]

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f) Function Value At the Right End-Point f ( b )   f : [ a, b ]  [  ,  ]. b  [ a, b ]  f ( b ) is defined at right end-point

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) x = c f ( c ) Function Value At An Internal Point f : [ a, b ]  [  ,  ]   c  [ a, b ]  f ( c ) is defined at an internal point

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R a n g e R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) x = c f ( c ) Function Values At Three Special Points f ( a ) f ( b ) f : [ a, b ]  [  ,  ]. a  [ a, b ]  f ( a ) is defined at left end-point b  [ a, b ]  f ( b ) is defined at right end-point c  [ a, b ]  f ( c ) is defined at an internal point  

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) Function Value Near Left End-Point   f : [ a, b ]  [  ,  ] a + h  [ a, b ]  f ( a + h ) is defined near left end-point on the right side a + h f ( a + h ) a – h  [ a, b ]  f ( a – h ) is NOT defined near left end-point on the left side

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) Function Value Near Right End-Point   f : [ a, b ]  [  ,  ] b + h  [ a, b ]  f ( b + h ) is NOT defined near right end- point on the right side b – h  [ a, b ]  f ( b – h ) is defined near right end-point on the left side b - h f ( b - h )

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) x = c f ( c ) c - h c + h Function Values Near An Internal Point f ( c - h ) f ( c  h ) f : [ a, b ]  [  ,  ]. c – h  [ a, b ]  f ( c – h ) is defined near an internal point on the left side c + h  [ a, b ]  f ( c + h ) is defined near an internal point on the right side   R a n g e

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R a n g e x-axis y-axis 0 a b Domain [ a , b ] Function ( f ) x = c f ( c ) c - h c + h Function Values Near An Internal Point f ( c - h ) f ( c  h ) f : [ a, b ]  [  ,  ].  

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We may also say that f ( x ) is the image under f of the pre-image x . The image set of the domain A is denoted by f (A). It is always necessary that f (A)  B . In case, i) f (A)  B, f is said to be into ii) f (A) = B, f is said to be onto. Pre-image and Image

Representations of Functions : 

Representations of Functions x a e f ( x ) f ( e ) f ( a ) 1 - 1 2 - 2 3 -3 Many - to - one (onto ) One -to- one 1 4 1 -1 2 -2 1. Arrow diagrams: One-to-one ( into) f f f A B A B A B

2. Numerical or Tabular Form: 

2. Numerical or Tabular Form Here the entries in the first column form the domain; and those in the second column form the range . x f ( x ) 2 3 5 1 -2 11 9 12 -7 9 The rule or correspondence f is: f (2) =11, f (3) = 9, f (5) = 12, f (1) = -7, f (-2) = 9 .

Function and Graph (a): 

Function and Graph (a) ( x , f ( x )) x f ( x ) x-axis y-axis O f 2. Graphical representation

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Symbolic or Algebraic form y = f ( x ). In particular, y = x, y = mx + c, v) Verbal statement or Description 1. v(t) is the velocity (instantaneous) velocity at time t. 2. A(r) is the area of a circle of radius r. 3. h(t) is the height of a balloon after time t. 4. T(t) temperature at time y = f ( x ) = x 3 … etc .

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Function in a Nutshell Sets : A = {-3, -1, 1, 3, 4}, B = {3, 2, 1, -2} Relation: { (-3, 3), (-3, 2), (-1, 2), (1, 1), (3, -2), (4, -2)} Function as a special kind of relation : Function as a graph : (-3, 3 ), (-1, 2 ), (1, 1 ), (3, -2 ), (4, -2 ) { } Function as table: x y -3 3 -1 2 1 1 3 -2 4 -2 Domain { -3, -1, 1, 3, 4 } Range { -2, 1, 2, 3 } Arrow diagram or Mapping: - 3 -1 1 3 4 3 2 1 -2 R

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A function is generally defined from a set to another. A beginner can have a better understanding of function and various properties of function by considering functions from an arbitrary set to the set of real numbers . Such functions are called real-valued functions. In many occasions, we may also have to restrict our discussion to real-valued function of a real variable . Real-valued Function

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Suppose we have a set of real number R. A function f : R → R which associates a number x  R with a unique number f ( x )  R, is called a real-valued function of a real variable . Some typical real valued functions are: Unit-valued function Identity function Constant function Absolute value function Greatest integer function Real-valued function of a real variable

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If x  R, a function f defined by f ( x ) = 1, is a real-valued function of a real variable. We call it a unit-valued function . Unit-valued function

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It is a horizontal line parallel to the x-axis at a unit distance . x-axis y-axis 0 y = f ( x ) = 1 Graph of Unit-valued function

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If x  R and a function f defined by f ( x ) = x , is another important real-valued function. This function is called an identity function . It associates or transforms a real number into the same real number. It is denoted by In particular, f (1) = 1, f (2) = 2, …, f ( c ) = c Identity Function

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Graph of Identity Function x-axis y-axis 0 y = f ( x ) = x

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The next function of our interest is the constant function. A constant function f is defined by f ( x ) = c , where c is a constant in R. Constant Function.

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Graph of Constant Function. It is a horizontal line parallel to the x-axis at a distance of c units x-axis y-axis 0 y = f ( x ) = c c

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Absolute Value Function. Let x denote any real number. The absolute value (or modulus or numerical value ) of x , written  x  , is a non-negative number defined by In particular,  - 3 = 3,  3  = 3,  0  = 0,  =  .

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The graph of the absolute value function is shown below: x-axis y-axis 0 y = f ( x ) =  x 

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Every one of us is quite familiar with “ stair-case ”. We can describe it as the real-valued function defined by f ( x ) = [ x ] , where [ x ] denotes the integer not greater than x itself . In particular, f (1.5) = [1.5] = 1, f ( 7/3) = [7/3] = 2 = f (5/2) , f (1/3) = 0, … The graph of this function looks like a stair-case and so this function is known as the stair-case function . Greatest Integer Function .

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Graph of the greatest integer function. x-axis y-axis 0 y = f ( x ) = [ x ] Stair-case function

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In mathematics, we often have to combine two or more functions. We now briefly mention how this can be done. For simplicity, we shall assume that f and g are two functions defined on subsets of the set of real numbers. Two functions f and g are said to be equal if and only if f and g are defined on the same set R and have same domain D  R and f ( a ) = g ( a ) for every a  R. Algebra of Real-valued functions

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Given any two functions f and g , Their i) sum , denoted by f + g ; ii) difference, denoted by f – g ; iii) product, denoted by ( f . g ); iv) quotient, denoted by ( f / g ) are defined by i) ( f + g )( x ) = f ( x ) + g ( x ) ii) ( f – g )( x ) = f ( x ) – g ( x ) iii) ( f . g )( x ) = f ( x ). g ( x ) iv) ( f / g )( x ) = f ( x ) / g ( x ). Algebra of Real-valued functions

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When one function f is added to or subtracted from or multiplied by or divided by another function g , the domain of the resulting function is not necessarily the whole of the domain of f and/or that of g . It will be the set of values of x common to both. In the case of the quotient, we have to exclude those values of x for which g ( x ) = 0. Something More

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Further to the above rules we have ( f + k )( x ) = f ( x ) + k , where k is a constant, (  x  ) ( x ) =  f ( x )  f n ( x ) = ( f ( x )) n ( kf )( x ) = ( kf )( x ) = f ( kx ). Something More

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Some Examples

Some Standard Types of Functions:: 

Some Standard Types of Functions: A function f : A → B is injective if for every x and y in A, x ≠ y implies f ( x ) ≠ f ( y ). a) Injection: Equivalently, a function f : A → B is injective if for every x and y in A, f ( x ) = f ( y ) implies x = y . For example, the function f : R → R defined by y = f ( x ) = x 3 is one-one or injective, since x 3 = y 3 implies x = y.

b) Surjection:: 

b) Surjection: A function f : A → B is surjective or onto, if for each in B, there is an x in A such that f ( x ) = y . y = f ( x ) = x 2 is surjective or onto , since (+ x ) 2 = y = (- x ) 2 . is NOT surjective or onto, since no negative number has a pre-image. For example, the function f : R  R + defined by But, the function f : R → R defined by y = f ( x ) = x 2

c) Bijection:: 

c) Bijection: A function f : A → B is bijective if it is both one-one and onto , i.e., if it is both injective and surjective . Example, the function f : R → R defined by y = f ( x ) = 2 x + 3 is both one-one and onto; and hence it is bijective.

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d) Extension and restriction Suppose A  B and C  D. Let two functions f and g be defined by g : A → C and f : B→ D. Then f is called an extension of g; and g is called a restriction of f .

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Let a function is defined by f : A → B . Then, given an element y  B, one or more values of x  A do exist . Inverse Image and Inverse Function a) Inverse image of an element f -1 ( y ) This set is often denoted by ( read f inverse of y ). In symbols, if a function f : A → B, then The set of all such values of x is known as the inverse image of the element x with respect to f . = { x . x  A , y = f ( x )}. f -1 ( y )

b) Inverse function: 

b) Inverse function If a function f : A→ B defined by y = f ( x ) is both one-to-one and onto (i.e. both injective and surjective) then the inverse function f -1 : B → A is defined by f -1 ( y ) = x

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D omain of f = Range of f - 1 Range of f = Domain of f - 1 Domain and Range of inverse function This is illustrated in the following figure:

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In particular , The function defined by y = f ( x ) = 2 x + 3 is both one-one and onto; And so its inverse f -1 exists; and is defined by f -1 ( y ) = ( y – 3)/2 . We often write such an inverse in the form f - 1 ( x ) = ( x – 3)/2 . ( Why ?)

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Inverse function computation scheme Suppose f : A → B is a given function Let us now see how we can find The scheme below illustrates the procedure : f -1

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Geometrical interpretation of inverse function If we draw the graphs of the function f and its inverse, we shall find each graph as a mirror image of the other, the mirror being placed along the line y = x. f -1

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In other words, The graph of is the reflection of the graph of f about the line y = x. Geometrical interpretation of inverse function f -1 That is to say, The graphs of f and are symmetric with respect to the line y = x.

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Its inverse : R → R is well defined by Let R denote the set of real numbers. Suppose we have a function f : R → R defined by y = f ( x ) =3 x – 2. Obviously, f is bijective. f -1 y = f -1 ( x ) =

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Their graphs are

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Consider the function f : A→B, where A = {1, 2, 3} and B = {1, 4, 9} defined by y = f ( x ) = x 2 . It is obviously bijective, and its inverse f -1 : B → A is well defined by y = f -1 ( x ) = + .

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. Their graphs are shown below:

Composition of functions: 

Composition of functions x ● gof = g ( f ) f g A B C f ( x ) ● ● g ( f ( x )) Let us first have a look at the following animation:

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Let f : A → B and g : B → C be two functions. Given x  A, a function f : A → B associates a unique element f ( x )  B If we further define a function g : B → C, then, the image of f ( x ) under g is g ( f ( x )) is unique and belongs to C. We thus find corresponding to an element of A a unique element of C via an element of B. That is, we have a new function from A to B This new function is known as the composite function of f and g

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This function is denoted by gof : A → C, is defined by z = g ( f ( x )). A composition is also known as a function of a function or product function . An important property of composition of functions is that it is associative. That is, If f : A → B , g : B → C and g : C → D , then ( hog ) of = ho ( gof ).

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Algebraic and Non-Algebraic Functions Plus Graphs

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. Trigonometric functions are functions defined on the set of angles. Traditionally, they are defined for angles of a right angled triangle. We often shall have to define them for angles of any magnitude also. . Trigonometric functions

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An angle is formed when two half-lines or rays starts from a common point called the vertex of the angle. The second way is to rotate a ray about its end-point or initial point . The angle is said to be positive if the rotation is anti-clockwise and negative if it is clockwise . Angle defined in this way may be of any magnitude. An angle is generally measured in degrees or radians . Trigonometric functions

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We agree that 1 revolution = 360 degrees or 2  radians. 1 degree = radians, or 1o = rads. and 1 radian = degrees, or, 1 rad = . Also, 1o = 60 minute = 60  60 seconds = 3600 seconds, or, 1 o = 60 = 3600  . An important relation connecting the central angle  (in radians) subtended by an arc length s at the centre of a circle of radius r is s = r  .

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This relation directly follows from the geometrical fact that “ Angles at the centre of a circle are proportional to the arcs on which they stand .”

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Trigonometric ratios of an acute angle θ

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Trigonometric ratios of any angle θ

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Trigonometric ratios of some special angles

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The exponential function f of a real variable x with base a ( a > 0) is defined by y = f ( x ) = a x . Exponential Function Here, a positive number a is raised to a real power x ; and hence its function value is always positive. This means that the graph of the exponential function will be located in the first and second quadrants only.

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Three typical cases: 0 < a < 1, b) a = 1, c) a > 1. Let us begin by closely examine the following animation : a = 1 0 < a < 1 a > 1

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Observation: 0 < a < 1: The function values gradually decreases with increasing x and approaches extremely close to 0; but never becomes zero. Its graph is very, very large on its left side. As the graph progresses to the right, it starts to move extremely closer and closer to the x -axis on the right; but not meeting the axis. b) a = 1: The function value is constant and is equal to 1. So, the graph is a straight line parallel to the x-axis at a unit distance.

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Observation: c) a > 1: The function values is very small on the left and is extremely close to the x-axis. As the graph progresses to the right, it starts to grow faster and faster and shoots off the top of the graph very quickly.

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The three typical cases are illustrated in the following diagram.

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The special case of the exponential function defined by y = f ( x ) = e x where e is an irrational number, approximately 2.71828183, designed after the 18th century Swiss mathematician, Leonhard Euler , is called the natural exponential function . The natural exponential function

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Certain common characteristics exponential graphs •  graph crosses the y -axis at (0,1) •  when a > 1, the graph increases •  when 0 < a < 1, the graph decreases •  the domain is all real numbers •  the range is all positive real numbers (never zero) •  graph passes the vertical line test - it is a function •  graph passes the horizontal line test - its inverse is also a function. •  graph is asymptotic to the x -axis - gets very, very close to the x -axis but does not touch it or cross Exponential functions are one-to-one functions

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Since the exponential function defined by y = f ( x ) = a x , ( a > 0) is one-to-one and onto ( or bijective), it has an inverse: and the inverse is also a function. This inverse function is called the logrithmic function. A logarithm is thus nothing but an exponent that is written in a special way. For example, in the exponential equation: 32 = 9, the base is 3 and the exponent is 2. We can write this equation in logarithm form (with identical meaning) as follows: log39 = 2 We say this as "the logarithm of 9 to the base 3 is 2". What we have effectively done is to move the exponent down on to the main line.

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In the exponential equation 3 2 = 9 the base is 3 and the exponent is 2. An alternate way of describing this equation is "the logarithm of 9 to the base 3 is 2". In symbols, we write this as Logarithm of a number to the base 10 is known as the common logarithm . Logarithm of a number to the base e is known as the natural logarithm .

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Since the exponential function defined by y = f ( x ) = a x , ( a > 0) is one-to-one and onto ( or bijective), it has an inverse: and the inverse is also a function. This inverse function is called the logrithmic function. Here is how we arrive at the inverse algebraically: set the equation equal to y • swap the x and y ( x > 0) • solve for y by rewriting in log form

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The fact that x > 0 and a > 0 the exponential function defined by y = f ( x ) = a x , ( a > 0) Tells us that the graph of this function lies in the first and fourth quadrants. .

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Logarithm functions, graphed for various bases: red is to base e , green is to base 10, and purple is to base 1.7

Domain of Logarithmic Functions: 

Domain of Logarithmic Functions Because the logarithmic function is the inverse of the exponential function, its domain and range are reversed. The domain is { x | x > 0 } and the range will be all real numbers.

Properties of Logarithms: 

Properties of Logarithms General Common Natural logarithm logarithm logarithm 1. log a 1 = 0 1. log 1 = 0 1. log e 1 = 0 = ln 1 2. log a a = 1 2. log 10 = 1 2. log e e = 1 = ln e 3. log a a x = 0 3. log 10 x = x 3. log e e x = x = ln x 4. a log a x = x 4. 10 log 10 x = x 4. e log e x = x = ln x

Reflection on the line y = x: 

Reflection on the line y = x

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End of Lecture ...

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GOOD LUCK EVERYBODY NEPAL MATHEMATICS CENTRE SEE YOU NEXT TIME THANKS