The document discusses different types of functions including:
1. A function maps each input to a single output and can be represented by f(x) = y.
2. Types of functions include one-to-one, onto, and bijective functions.
3. Important concepts are the domain, range, and inverse of a function.
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.
Gives idea about function, one to one function, inverse function, which functions are invertible, how to invert a function and application of inverse functions.
Convex Analysis and Duality (based on "Functional Analysis and Optimization" ...Katsuya Ito
In this presentation, we explain the monograph ”Functional Analysis and Optimization” by Kazufumi Ito
https://kito.wordpress.ncsu.edu/files/2018/04/funa3.pdf
Our goal in this presentation is to
-Understand the basic notions of functional analysis
lower-semicontinuous, subdifferential, conjugate functional
- Understand the formulation of duality problem
primal (P), perturbed (Py), and dual (P∗) problem
-Understand the primal-dual relationships
inf(P)≤sup(P∗), inf(P) = sup(P∗), inf supL≤sup inf L
We will define what is a function “formally”, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
(a) Natural Numbers : N = {1,2,3,4,...}
(b) Whole Numbers : W = {0,1,2,3,4, }
(c) Integer Numbers :
or Z = {...–3,–2,–1, 0,1,2,3, },
Z+ = {1,2,3,....}, Z– = {–1,–2,–3, }
Z0 = {± 1, ± 2, ± 3, }
(d) Rational Numbers :
p
Q = { q ; p, q z, q 0 }
(i) R0 : all real numbers except 0 (Zero).
(j) Imaginary Numbers : C = {i,, }
(k) Prime Numbers :
These are the natural numbers greater than 1 which is divisible by 1 and itself only, called prime numbers.
Ex. 2,3,5,7,11,13,17,19,23,29,31,37,41,...
(l) Even Numbers : E = {0,2,4,6, }
(m) Odd Numbers : O = {1,3,5,7, }
Ex. {1,
Note :
5
, –10, 105,
3
22 20
7 , 3
, 0 ....}
The set of the numbers between any two real numbers is called interval.
(a) Close Interval :
(i) In rational numbers the digits are repeated after decimal.
(ii) 0 (zero) is a rational number.
(e) Irrational numbers: The numbers which are not rational or which can not be written in the form of p/q ,called irrational numbers
Ex. { , ,21/3, 51/4, ,e, }
Note:
(i) In irrational numbers, digits are not repeated after decimal.
(ii) and e are called special irrational quantities.
(iii) is neither a rational number nor a irrational number.
(f) Real Numbers : {x, where x is rational and irrational number}
20
[a,b] = { x, a x b }
(b) Open Interval:
(a, b) or ]a, b[ = { x, a < x < b }
(c) Semi open or semi close interval:
[a,b[ or [a,b) = {x; a x < b}
]a,b] or (a,b] = {x ; a < x b}
Let A and B be two given sets and if each element a A is associated with a unique element b B under a rule f , then this relation is called function.
Here b, is called the image of a and a is called the pre- image of b under f.
Note :
(i) Every element of A should be associated with
Ex. R = { 1,1000, 20/6, ,
, –10, –
,.....}
3
B but vice-versa is not essential.
(g) Positive Real Numbers: R+ = (0,)
(h) Negative Real Numbers : R– = (– ,0)
(ii) Every element of A should be associated with a unique (one and only one) element of but
any element of B can have two or more rela- tions in A.
3.1 Representation of Function :
It can be done by three methods :
(a) By Mapping
(b) By Algebraic Method
(c) In the form of Ordered pairs
(A) Mapping :
It shows the graphical aspect of the relation of the elements of A with the elements of B .
Ex. f1:
f2 :
f3 :
f4 :
In the above given mappings rule f1 and f2
shows a function because each element of A is
associated with a unique element of B. Whereas
f3 and f4 are not function because in f 3, element c is associated with two elements of B, and in f4 , b is not associated with any element
of B, which do not follow the definition of function. In f2, c and d are associated with same element, still it obeys the rule of definition of function because it does not tell that every element of A should be associated with different elements of B.
(B) Algebraic Method :
It shows the relation between the elem
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
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Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
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1. 16 CIVIL 15/07/16 1
INTRODUCTION TO FUNCTIONSINTRODUCTION TO FUNCTIONS
The term function was recognized by a GermanThe term function was recognized by a German
Mathematician LEIBNIZ .Mathematician LEIBNIZ .
Why………to describe the dependence of oneWhy………to describe the dependence of one
quantity to an other quantity.quantity to an other quantity.
DefinitionDefinition:1. A:1. A functionfunction is a relation thatis a relation that
gives agives a single outputsingle output number for every validnumber for every valid
input number (x values cannot be repeated)input number (x values cannot be repeated)
2. Function is input and out put device.2. Function is input and out put device.
3.A function is like machine that assigns a3.A function is like machine that assigns a
unique output to every allowable input.unique output to every allowable input.
2. 16 CIVIL 2
FunctionsFunctions
4.A4.A functionfunction f from a set X to a set Y is anf from a set X to a set Y is an
assignmentassignment of exactly one element of Y to eachof exactly one element of Y to each
element of X.element of X.
We writeWe write
f(x) = y or y = f(x)f(x) = y or y = f(x)
if y is the unique element of B assigned by theif y is the unique element of B assigned by the
function f to the element x of A.function f to the element x of A.
If f is a function from A to B, we writeIf f is a function from A to B, we write
f: Af: A→→BB
(note: Here, “(note: Here, “→→“ has nothing to do with if… then)“ has nothing to do with if… then)
3. 16 CIVIL 3
FunctionsFunctions
5.Function is rule to which assigns a value of5.Function is rule to which assigns a value of
independent variable which corresponds to uniqueindependent variable which corresponds to unique
value of dependent variable.value of dependent variable.
If f:AIf f:A→→B, we say that A is theB, we say that A is the domaindomain of f and Bof f and B
is theis the co domainco domain of f.of f.
If f(x) = y, we say that y is theIf f(x) = y, we say that y is the imageimage of x and x isof x and x is
thethe pre-imagepre-image of y.of y.
TheThe rangerange of f:Aof f:A→→B is the set of all images ofB is the set of all images of
elements of A.elements of A.
We say that f:AWe say that f:A→→BB mapsmaps A to B.A to B.
4. 16 CIVIL 4
FunctionFunction
A function is a rule that maps a number toA function is a rule that maps a number to
another unique number.The input to theanother unique number.The input to the
function is called the independent variable, andfunction is called the independent variable, and
is also called the argument of the function. Theis also called the argument of the function. The
output of the function is called the dependentoutput of the function is called the dependent
variable.variable.
A Swiss mathematician Leon-Hard Euler inventedA Swiss mathematician Leon-Hard Euler invented
a symbolic way to write statement y is functiona symbolic way to write statement y is function
of x as y = f(x) read as y is equal to f of xof x as y = f(x) read as y is equal to f of x
Example:Example:
y = xy = x + 1+ 1
5. Function
FunctionFunction - for every x there is exactly one y.- for every x there is exactly one y.
DomainDomain - set of x-values- set of x-values
RangeRange - set of y-values- set of y-values
6. 16 CIVIL 6
Types of FunctionsTypes of Functions
A function f:AA function f:A→→B is said to beB is said to be one-to-oneone-to-one (or(or
injectiveinjective), if and only if), if and only if
∀∀x, yx, y ∈∈ A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y)
In other words:In other words: f is one-to-one if and only if itf is one-to-one if and only if it
does not map two distinct elements of A onto thedoes not map two distinct elements of A onto the
same element of B. orsame element of B. or
Distinct elements of A have distinct imagesDistinct elements of A have distinct images
7. 16 CIVIL 7
Types of FunctionsTypes of Functions
Example:Example:
f(Linda) = Moscowf(Linda) = Moscow
f(Max) = Bostonf(Max) = Boston
f(Kathy) = Hong Kongf(Kathy) = Hong Kong
f(Peter) = Bostonf(Peter) = Boston
Is f one-to-one?Is f one-to-one?
No, Max and Peter areNo, Max and Peter are
mapped onto the samemapped onto the same
element of the image.element of the image.
g(Linda) = Moscowg(Linda) = Moscow
g(Max) = Bostong(Max) = Boston
g(Kathy) = Hong Kongg(Kathy) = Hong Kong
g(Peter) = New Yorkg(Peter) = New York
Is g one-to-one?Is g one-to-one?
Yes, each element isYes, each element is
assigned a uniqueassigned a unique
element of the image.element of the image.
8. 16 CIVIL 8
Types of FunctionsTypes of Functions
How can we prove that a function f is one-to-one?How can we prove that a function f is one-to-one?
Whenever you want to prove something, first takeWhenever you want to prove something, first take
a look at the relevant definition(s):a look at the relevant definition(s):
∀∀x, yx, y∈∈A (f(x) = f(y)A (f(x) = f(y) →→ x = y)x = y)
Example:Example:
f:f:RR→→RR
f(x) = xf(x) = x22
Disproof by counterexample:
f(3) = f(-3), but 3f(3) = f(-3), but 3 ≠≠ -3, so f is not one-to-one.-3, so f is not one-to-one.
9. 9
Types of FunctionsTypes of Functions
A function f:AA function f:A→→B is calledB is called ontoonto, or, or surjectivesurjective, if, if
and only if for every element yand only if for every element y ∈∈ B there is anB there is an
element xelement x ∈∈ A with f(x) = yA with f(x) = y
In other words, f is onto if and only if itsIn other words, f is onto if and only if its rangerange isis
itsits entire co domainentire co domain. e.g.. e.g.
A function f: AA function f: A→→B is aB is a one-to-one correspondenceone-to-one correspondence,,
or aor a bijectionbijection, if and only if it is both one-to-one, if and only if it is both one-to-one
and onto.and onto.
Obviously, if f is a bijection and A and B are finiteObviously, if f is a bijection and A and B are finite
sets, then |A| = |B|.sets, then |A| = |B|.
3
y x=
10. 10
InversionInversion
An interesting property of bijection is thatAn interesting property of bijection is that
they have anthey have an inverse functioninverse function..
TheThe inverse functioninverse function of the bijection f:Aof the bijection f:A→→BB
is the function fis the function f-1-1
:B:B→→A withA with
ff-1-1
(y) = x whenever f(x) = y.(y) = x whenever f(x) = y.
11. 11
InversionInversion
Example:Example:
f(Linda) = Moscowf(Linda) = Moscow
f(Max) = Bostonf(Max) = Boston
f(Kathy) = Hong Kongf(Kathy) = Hong Kong
f(Peter) = Lf(Peter) = Lüübeckbeck
f(Helena) = New Yorkf(Helena) = New York
Clearly, f is bijective.Clearly, f is bijective.
The inverse functionThe inverse function
ff-1-1
is given by:is given by:
ff-1-1
(Moscow) = Linda(Moscow) = Linda
ff-1-1
(Boston) = Max(Boston) = Max
ff-1-1
(Hong Kong) = Kathy(Hong Kong) = Kathy
ff-1-1
(L(Lüübeck) = Peterbeck) = Peter
ff-1-1
(New York) = Helena(New York) = Helena
Inversion is onlyInversion is only
possible for bijectionspossible for bijections
(= invertible functions)(= invertible functions)
12. Types of FunctionTypes of Function
Constant Function:Constant Function:
Let ‘A’ and ‘B’ be any two non–emptyLet ‘A’ and ‘B’ be any two non–empty
sets, then a function ‘ff’ from ‘A’ tosets, then a function ‘ff’ from ‘A’ to
‘B’ is called Constant Function if and‘B’ is called Constant Function if and
only if range of ‘f’ is a singleton.only if range of ‘f’ is a singleton.
Algebraic Function: TheAlgebraic Function: The functionfunction
defined by algebraic expression aredefined by algebraic expression are
called algebraic function.called algebraic function.
12
13. FunctionsFunctions
aann is called theis called the leading coefficientleading coefficient
nn is theis the degreedegree of the polynomialof the polynomial
aa00 is called theis called the constant termconstant term
Polynomial FunctionPolynomial Function
AA polynomial function of degreepolynomial function of degree nn in the variablein the variable xx isis
a function defined bya function defined by
where eachwhere each aaii is real,is real, aann 0, and 0, and nn is a whole number.is a whole number.
01
1
1)( axaxaxaxP n
n
n
n ++++= −
−
14. Polynomial FunctionsPolynomial Functions
The largest exponent within theThe largest exponent within the
polynomial determines the degree of thepolynomial determines the degree of the
polynomial.polynomial.
Polynomial
Function in
General Form
Degree
Name of
Function
1 Linear
2 Quadratic
3 Cubic
4 Quarticedxcxbxaxy ++++= 234
dcxbxaxy +++= 23
cbxaxy ++= 2
baxy +=
15. Even and Odd FunctionsEven and Odd Functions
A function is y = f(x) isA function is y = f(x) is eveneven if, for each x in theif, for each x in the
domain of f,domain of f,
f(-x) = f(x)f(-x) = f(x)
A function is y = f(x) isA function is y = f(x) is oddodd if, for each x in theif, for each x in the
domain of f,domain of f,
f(-x) = -f(x)f(-x) = -f(x)
An even function is symmetric about the y-axis.An even function is symmetric about the y-axis.
An odd function is symmetric about the origin.An odd function is symmetric about the origin.
16. Ex. g(x) = xEx. g(x) = x33
- x- x
g(-x) = (-x)g(-x) = (-x)33
– (-x) = -x– (-x) = -x33
+ x =+ x =-(x-(x33
– x)– x)
Therefore, g(x) is odd because f(-x) = -f(x)Therefore, g(x) is odd because f(-x) = -f(x)
Ex. h(x) = xEx. h(x) = x22
+ 1+ 1
h(-x) = (-x)h(-x) = (-x)22
+ 1 = x+ 1 = x22
+ 1+ 1
h(x) is even because f(-x) = f(x)h(x) is even because f(-x) = f(x)
17. 17
CompositionComposition
TheThe compositioncomposition of two functions g:Aof two functions g:A→→B andB and
f:Bf:B→→C, denoted by fC, denoted by f°°g, is defined byg, is defined by
(f(f°°g)(a) = f(g(a))g)(a) = f(g(a))
This means thatThis means that
• firstfirst, function g is applied to element a, function g is applied to element a∈∈A,A,
mapping it onto an element of B,mapping it onto an element of B,
• thenthen, function f is applied to this element of, function f is applied to this element of
B, mapping it onto an element of C.B, mapping it onto an element of C.
• ThereforeTherefore, the composite function maps, the composite function maps
from A to C.from A to C.
19. 19
CompositionComposition
Composition of a function and its inverse:Composition of a function and its inverse:
(f(f-1-1
°°f)(x) = ff)(x) = f-1-1
(f(x)) = x(f(x)) = x
The composition of a function and its inverseThe composition of a function and its inverse
is theis the identity functionidentity function i(x) = x.i(x) = x.
20. 20
Floor and Ceiling FunctionsFloor and Ceiling Functions
TheThe floorfloor andand ceilingceiling functions map the realfunctions map the real
numbers onto the integers (numbers onto the integers (RR→→ZZ).).
TheThe floorfloor function assigns to rfunction assigns to r∈∈RR the largestthe largest
zz∈∈ZZ with zwith z ≤≤ r, denoted byr, denoted by rr..
Examples:Examples: 2.32.3 = 2,= 2, 22 = 2,= 2, 0.50.5 = 0,= 0, -3.5-3.5 = -4= -4
TheThe ceilingceiling function assigns to rfunction assigns to r∈∈RR the smallestthe smallest
zz∈∈ZZ with zwith z ≥≥ r, denoted byr, denoted by rr..
Examples:Examples: 2.32.3 = 3,= 3, 22 = 2,= 2, 0.50.5 = 1,= 1, -3.5-3.5 = -3= -3