In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function (a higher-order function). Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.
Functional details
Duality
The mapping
is a function, where x0 is an argument of a function f.
At the same time, the mapping of a function to the value of the function at a point
Provided that f is a linear function from a linear vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.
Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
The Junior Certificate (Irish:Teastas Sóisearach) is an educational qualification awarded in Ireland by the Department of Education and Skills to students who have successfully completed the junior cycle of secondary education, and achieved a minimum standard in their Junior Certification examinations. These exams, like those for the Leaving Certificate, are supervised by the State Examinations Commission.
A "recognised pupil"<ref name"">Definitions, Rules and Programme for Secondary Education, Department of Education, Ireland, 2004</ref> who commences the Junior Cycle must reach at least 12 years of age on 1 January of the school year of admission and must have completed primary education; the examination is normally taken after three years' study in a secondary school. Typically a student takes 9 to 13 subjects – including English, Irish and Mathematics – as part of the Junior Cycle. The examination does not reach the standards for college or university entrance; instead a school leaver in Ireland will typically take the Leaving Certificate Examination two or three years after completion of the Junior Certificate to reach that standard.
In the context of the C++ programming language, functional refers to a header file that is part of the C++ Standard Library and provides a number of predefined class templates for function objects, including arithmetic operations, comparisons, and logical operations. Instances of these class templates are C++ classes that define a function call operator, and the instances of these classes can be called as if they were functions. It is possible to perform very sophisticated operations without actually writing a new function object, simply by combining predefined function objects and function object adaptors.
The class template std::function provided by C++11 is a general-purpose polymorphicfunction wrapper. Instances of std::function can store, copy, and invoke any callable target—functions, lambda expressions (expressions defining anonymous functions), bind expressions (instances of function adapters that transform functions to other functions of smaller arity by providing values for some of the arguments), or other function objects.
In computer science, functional programming is a programming paradigm—a style of building the structure and elements of computer programs—that treats computation as the evaluation of mathematical functions and avoids changing-state and mutable data. It is a declarative programming paradigm, which means programming is done with expressions. In functional code, the output value of a function depends only on the arguments that are input to the function, so calling a function f twice with the same value for an argument x will produce the same result f(x) each time. Eliminating side effects, i.e. changes in state that do not depend on the function inputs, can make it much easier to understand and predict the behavior of a program, which is one of the key motivations for the development of functional programming.
Learning about functions is critical in math, especially in Algebra. Many students struggle with the concept of what a function is and how to determine is a relation is a function. This video will explain in detail the foundation of what a function is in algebra to include how to evaluate a function and what the domain and range represent with respect to functions.
TabletClass Math Algebra:
https://tabletclass-academy.teachable.com/p/tabletclass-math-algebra1
published: 05 Mar 2021
06 - What is a Function in Math? (Learn Function Definition, Domain & Range in Algebra)
Get more lessons like this at http://www.MathTutorDVD.com.
Here you will learn what a function is in math, the definition of a function, and why they are important.
Functions have applications in algebra, calculus, science, and engineering. We first begin by describing a function as a mathematical machine that takes input numbers "x" and computes output numbers "f(x)".
We then describe the input numbers "x" as being the domain of the function, while the output numbers f(x) are the range of the function. We will plot several functions including linear functions, quadratic functions, cubic functions, and more.
published: 19 Dec 2018
Introduction to Function and Types of Function - Functions - Discrete Mathematics
Subject - Discrete Mathematics
Video Name - Introduction to Function and Types of Function
Chapter - Functions
Faculty - Prof. Farhan Meer
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded and IOT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Dis...
published: 09 Apr 2022
Functional (mathematics)
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Functional (mathematics)
In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers.In other words, it is a function that takes a vector as its input argument, and returns a scalar.
=======Image-Copyright-Info=======
Image is in public domain
Author-Info: Dariusofthedark at English Wikipedia
Image Source: https://en.wikipedia.org/wiki/File:Arclength.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://ww...
published: 22 Jan 2016
What is a function? | Functions and their graphs | Algebra II | Khan Academy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function
Watch the next lesson: https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/difference-between-equations-and-functions?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Missed the previous lesson?
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/non-linear-systems-tutorial/v/systems-of-nonlinear-equations-3?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Algebra II on Khan Academy: Your studies in algebra 1 have built a solid foundation from which you can explore linear equations, inequalities, and functions. In alge...
published: 06 Jun 2013
Functional Equation JEE Main PYQ 🔥|JEE Main and Advanced |Dr. Vipin Jaiswal
Functional Equation JEE Main PYQ 🔥|JEE Main and Advanced |Dr. Vipin Jaiswal
#jee
#iitjee
#jeemains2025
#jeeadvanced2025
#jeepyq
#maths
#jeemaths
#mathstricks
#mathshortcut
#physicswallah
#unacademy
#allen
published: 05 Jan 2025
Composite Functions
This algebra video tutorial provides a basic introduction into composite functions. it explains how to evaluate composite functions. This video contains a few examples and practice problems.
Functions - Free Formula Sheet: https://bit.ly/3Ai8NZv
__________________________________
Composite Functions:
https://www.youtube.com/watch?v=ZFPkQkURSxk
How To Evaluate Composite Functions:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Composite Functions - Data Table:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Comp. of 3 Functions - Data Table:
https://www.youtube.com/watch?v=Ote1KvW6BXw
Decomposing Functions:
https://www.youtube.com/watch?v=NmfaC7etZms
_____________________________________
Increasing and...
Learning about functions is critical in math, especially in Algebra. Many students struggle with the concept of what a function is and how to determine is a re...
Learning about functions is critical in math, especially in Algebra. Many students struggle with the concept of what a function is and how to determine is a relation is a function. This video will explain in detail the foundation of what a function is in algebra to include how to evaluate a function and what the domain and range represent with respect to functions.
TabletClass Math Algebra:
https://tabletclass-academy.teachable.com/p/tabletclass-math-algebra1
Learning about functions is critical in math, especially in Algebra. Many students struggle with the concept of what a function is and how to determine is a relation is a function. This video will explain in detail the foundation of what a function is in algebra to include how to evaluate a function and what the domain and range represent with respect to functions.
TabletClass Math Algebra:
https://tabletclass-academy.teachable.com/p/tabletclass-math-algebra1
Get more lessons like this at http://www.MathTutorDVD.com.
Here you will learn what a function is in math, the definition of a function, and why they are import...
Get more lessons like this at http://www.MathTutorDVD.com.
Here you will learn what a function is in math, the definition of a function, and why they are important.
Functions have applications in algebra, calculus, science, and engineering. We first begin by describing a function as a mathematical machine that takes input numbers "x" and computes output numbers "f(x)".
We then describe the input numbers "x" as being the domain of the function, while the output numbers f(x) are the range of the function. We will plot several functions including linear functions, quadratic functions, cubic functions, and more.
Get more lessons like this at http://www.MathTutorDVD.com.
Here you will learn what a function is in math, the definition of a function, and why they are important.
Functions have applications in algebra, calculus, science, and engineering. We first begin by describing a function as a mathematical machine that takes input numbers "x" and computes output numbers "f(x)".
We then describe the input numbers "x" as being the domain of the function, while the output numbers f(x) are the range of the function. We will plot several functions including linear functions, quadratic functions, cubic functions, and more.
Subject - Discrete Mathematics
Video Name - Introduction to Function and Types of Function
Chapter - Functions
Faculty - Prof. Farhan Meer
Upskill and get P...
Subject - Discrete Mathematics
Video Name - Introduction to Function and Types of Function
Chapter - Functions
Faculty - Prof. Farhan Meer
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded and IOT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Discrete Mathematics - https://youtube.com/playlist?list=PLm_MSClsnwm9r6GqZgSn1fRRbPB7O-0rq
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#Functions #DiscreteMathematics
Subject - Discrete Mathematics
Video Name - Introduction to Function and Types of Function
Chapter - Functions
Faculty - Prof. Farhan Meer
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded and IOT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Discrete Mathematics - https://youtube.com/playlist?list=PLm_MSClsnwm9r6GqZgSn1fRRbPB7O-0rq
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#Functions #DiscreteMathematics
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Functional (mathematics)
In mat...
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Functional (mathematics)
In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers.In other words, it is a function that takes a vector as its input argument, and returns a scalar.
=======Image-Copyright-Info=======
Image is in public domain
Author-Info: Dariusofthedark at English Wikipedia
Image Source: https://en.wikipedia.org/wiki/File:Arclength.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=aAKcMzw4rMY
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Functional (mathematics)
In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers.In other words, it is a function that takes a vector as its input argument, and returns a scalar.
=======Image-Copyright-Info=======
Image is in public domain
Author-Info: Dariusofthedark at English Wikipedia
Image Source: https://en.wikipedia.org/wiki/File:Arclength.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=aAKcMzw4rMY
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functio...
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function
Watch the next lesson: https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/difference-between-equations-and-functions?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Missed the previous lesson?
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/non-linear-systems-tutorial/v/systems-of-nonlinear-equations-3?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Algebra II on Khan Academy: Your studies in algebra 1 have built a solid foundation from which you can explore linear equations, inequalities, and functions. In algebra 2 we build upon that foundation and not only extend our knowledge of algebra 1, but slowly become capable of tackling the BIG questions of the universe. We'll again touch on systems of equations, inequalities, and functions...but we'll also address exponential and logarithmic functions, logarithms, imaginary and complex numbers, conic sections, and matrices. Don't let these big words intimidate you. We're on this journey with you!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Algebra II channel:
https://www.youtube.com/channel/UCsCA3_VozRtgUT7wWC1uZDg?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function
Watch the next lesson: https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/difference-between-equations-and-functions?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Missed the previous lesson?
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/non-linear-systems-tutorial/v/systems-of-nonlinear-equations-3?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Algebra II on Khan Academy: Your studies in algebra 1 have built a solid foundation from which you can explore linear equations, inequalities, and functions. In algebra 2 we build upon that foundation and not only extend our knowledge of algebra 1, but slowly become capable of tackling the BIG questions of the universe. We'll again touch on systems of equations, inequalities, and functions...but we'll also address exponential and logarithmic functions, logarithms, imaginary and complex numbers, conic sections, and matrices. Don't let these big words intimidate you. We're on this journey with you!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Algebra II channel:
https://www.youtube.com/channel/UCsCA3_VozRtgUT7wWC1uZDg?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
This algebra video tutorial provides a basic introduction into composite functions. it explains how to evaluate composite functions. This video contains a few...
This algebra video tutorial provides a basic introduction into composite functions. it explains how to evaluate composite functions. This video contains a few examples and practice problems.
Functions - Free Formula Sheet: https://bit.ly/3Ai8NZv
__________________________________
Composite Functions:
https://www.youtube.com/watch?v=ZFPkQkURSxk
How To Evaluate Composite Functions:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Composite Functions - Data Table:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Comp. of 3 Functions - Data Table:
https://www.youtube.com/watch?v=Ote1KvW6BXw
Decomposing Functions:
https://www.youtube.com/watch?v=NmfaC7etZms
_____________________________________
Increasing and Decreasing Functions:
https://www.youtube.com/watch?v=Dyl7jPlJXOM
Relative Maximum & Minimum Values:
https://www.youtube.com/watch?v=GaltHJF6WyI
Domain and Range from a Graph:
https://www.youtube.com/watch?v=KirGQOwjBVI
Average Rate of Change of a Function:
https://www.youtube.com/watch?v=f_Q0qlwlclQ
The Difference Quotient of a Function:
https://www.youtube.com/watch?v=qQgVomi8lCc
_____________________________________
Evaluating Piecewise Functions:
https://www.youtube.com/watch?v=OYOXMyFKotc
Graphing Piecewise Functions:
https://www.youtube.com/watch?v=Uzw9tsGq2Pw
How To Find the Inverse of a Function:
https://www.youtube.com/watch?v=2zeYEx4eTdc
Verifying Inverse Functions:
https://www.youtube.com/watch?v=6QamgSPprjA
______________________________________
Horizontal Line Test - 1 to 1 Functions:
https://www.youtube.com/watch?v=6u9Qh1Vi_Qw
Vertical Line Test:
https://www.youtube.com/watch?v=Mxe2lX1htNk
Graphing Inverse Functions:
https://www.youtube.com/watch?v=ukEtad_aml4
Composite and Inverse Functions:
https://www.youtube.com/watch?v=GEgFmxxOJno
_______________________________________
Even and Odd Functions:
https://www.youtube.com/watch?v=fKyBOLsqRlo
Transformations of Functions:
https://www.youtube.com/watch?v=Tmdrjs9xufc
Linear Functions:
https://www.youtube.com/watch?v=BtcKotD6Ni8
Functions and Graphs - Problems:
https://www.youtube.com/watch?v=kvU9sOzT2mk
Functions - Test Review:
https://www.youtube.com/watch?v=1xATmTI-YY8
______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
This algebra video tutorial provides a basic introduction into composite functions. it explains how to evaluate composite functions. This video contains a few examples and practice problems.
Functions - Free Formula Sheet: https://bit.ly/3Ai8NZv
__________________________________
Composite Functions:
https://www.youtube.com/watch?v=ZFPkQkURSxk
How To Evaluate Composite Functions:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Composite Functions - Data Table:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Comp. of 3 Functions - Data Table:
https://www.youtube.com/watch?v=Ote1KvW6BXw
Decomposing Functions:
https://www.youtube.com/watch?v=NmfaC7etZms
_____________________________________
Increasing and Decreasing Functions:
https://www.youtube.com/watch?v=Dyl7jPlJXOM
Relative Maximum & Minimum Values:
https://www.youtube.com/watch?v=GaltHJF6WyI
Domain and Range from a Graph:
https://www.youtube.com/watch?v=KirGQOwjBVI
Average Rate of Change of a Function:
https://www.youtube.com/watch?v=f_Q0qlwlclQ
The Difference Quotient of a Function:
https://www.youtube.com/watch?v=qQgVomi8lCc
_____________________________________
Evaluating Piecewise Functions:
https://www.youtube.com/watch?v=OYOXMyFKotc
Graphing Piecewise Functions:
https://www.youtube.com/watch?v=Uzw9tsGq2Pw
How To Find the Inverse of a Function:
https://www.youtube.com/watch?v=2zeYEx4eTdc
Verifying Inverse Functions:
https://www.youtube.com/watch?v=6QamgSPprjA
______________________________________
Horizontal Line Test - 1 to 1 Functions:
https://www.youtube.com/watch?v=6u9Qh1Vi_Qw
Vertical Line Test:
https://www.youtube.com/watch?v=Mxe2lX1htNk
Graphing Inverse Functions:
https://www.youtube.com/watch?v=ukEtad_aml4
Composite and Inverse Functions:
https://www.youtube.com/watch?v=GEgFmxxOJno
_______________________________________
Even and Odd Functions:
https://www.youtube.com/watch?v=fKyBOLsqRlo
Transformations of Functions:
https://www.youtube.com/watch?v=Tmdrjs9xufc
Linear Functions:
https://www.youtube.com/watch?v=BtcKotD6Ni8
Functions and Graphs - Problems:
https://www.youtube.com/watch?v=kvU9sOzT2mk
Functions - Test Review:
https://www.youtube.com/watch?v=1xATmTI-YY8
______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
Learning about functions is critical in math, especially in Algebra. Many students struggle with the concept of what a function is and how to determine is a relation is a function. This video will explain in detail the foundation of what a function is in algebra to include how to evaluate a function and what the domain and range represent with respect to functions.
TabletClass Math Algebra:
https://tabletclass-academy.teachable.com/p/tabletclass-math-algebra1
Get more lessons like this at http://www.MathTutorDVD.com.
Here you will learn what a function is in math, the definition of a function, and why they are important.
Functions have applications in algebra, calculus, science, and engineering. We first begin by describing a function as a mathematical machine that takes input numbers "x" and computes output numbers "f(x)".
We then describe the input numbers "x" as being the domain of the function, while the output numbers f(x) are the range of the function. We will plot several functions including linear functions, quadratic functions, cubic functions, and more.
Subject - Discrete Mathematics
Video Name - Introduction to Function and Types of Function
Chapter - Functions
Faculty - Prof. Farhan Meer
Upskill and get Placements with Ekeeda Career Tracks
Data Science - https://ekeeda.com/career-track/data-scientist
Software Development Engineer - https://ekeeda.com/career-track/software-development-engineer
Embedded and IOT Engineer - https://ekeeda.com/career-track/embedded-and-iot-engineer
Get FREE Trial for GATE 2023 Exam with Ekeeda GATE - 20000+ Lectures & Notes, strategy, updates, and notifications which will help you to crack your GATE exam.
https://ekeeda.com/catalog/competitive-exam
Coupon Code - EKGATE
Get Free Notes of All Engineering Subjects & Technology
https://ekeeda.com/digital-library
Access the Complete Playlist of Subject Discrete Mathematics - https://youtube.com/playlist?list=PLm_MSClsnwm9r6GqZgSn1fRRbPB7O-0rq
Happy Learning
Social Links:
https://www.instagram.com/ekeeda_official/
https://in.linkedin.com/company/ekeeda.com
#Functions #DiscreteMathematics
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Functional (mathematics)
In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers.In other words, it is a function that takes a vector as its input argument, and returns a scalar.
=======Image-Copyright-Info=======
Image is in public domain
Author-Info: Dariusofthedark at English Wikipedia
Image Source: https://en.wikipedia.org/wiki/File:Arclength.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=aAKcMzw4rMY
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions/x2f8bb11595b61c86:evaluating-functions/v/what-is-a-function
Watch the next lesson: https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/difference-between-equations-and-functions?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Missed the previous lesson?
https://www.khanacademy.org/math/algebra2/systems_eq_ineq/non-linear-systems-tutorial/v/systems-of-nonlinear-equations-3?utm_source=YT&utm_medium=Desc&utm_campaign=AlgebraII
Algebra II on Khan Academy: Your studies in algebra 1 have built a solid foundation from which you can explore linear equations, inequalities, and functions. In algebra 2 we build upon that foundation and not only extend our knowledge of algebra 1, but slowly become capable of tackling the BIG questions of the universe. We'll again touch on systems of equations, inequalities, and functions...but we'll also address exponential and logarithmic functions, logarithms, imaginary and complex numbers, conic sections, and matrices. Don't let these big words intimidate you. We're on this journey with you!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Algebra II channel:
https://www.youtube.com/channel/UCsCA3_VozRtgUT7wWC1uZDg?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
This algebra video tutorial provides a basic introduction into composite functions. it explains how to evaluate composite functions. This video contains a few examples and practice problems.
Functions - Free Formula Sheet: https://bit.ly/3Ai8NZv
__________________________________
Composite Functions:
https://www.youtube.com/watch?v=ZFPkQkURSxk
How To Evaluate Composite Functions:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Composite Functions - Data Table:
https://www.youtube.com/watch?v=6vQxIG4lJ3w
Comp. of 3 Functions - Data Table:
https://www.youtube.com/watch?v=Ote1KvW6BXw
Decomposing Functions:
https://www.youtube.com/watch?v=NmfaC7etZms
_____________________________________
Increasing and Decreasing Functions:
https://www.youtube.com/watch?v=Dyl7jPlJXOM
Relative Maximum & Minimum Values:
https://www.youtube.com/watch?v=GaltHJF6WyI
Domain and Range from a Graph:
https://www.youtube.com/watch?v=KirGQOwjBVI
Average Rate of Change of a Function:
https://www.youtube.com/watch?v=f_Q0qlwlclQ
The Difference Quotient of a Function:
https://www.youtube.com/watch?v=qQgVomi8lCc
_____________________________________
Evaluating Piecewise Functions:
https://www.youtube.com/watch?v=OYOXMyFKotc
Graphing Piecewise Functions:
https://www.youtube.com/watch?v=Uzw9tsGq2Pw
How To Find the Inverse of a Function:
https://www.youtube.com/watch?v=2zeYEx4eTdc
Verifying Inverse Functions:
https://www.youtube.com/watch?v=6QamgSPprjA
______________________________________
Horizontal Line Test - 1 to 1 Functions:
https://www.youtube.com/watch?v=6u9Qh1Vi_Qw
Vertical Line Test:
https://www.youtube.com/watch?v=Mxe2lX1htNk
Graphing Inverse Functions:
https://www.youtube.com/watch?v=ukEtad_aml4
Composite and Inverse Functions:
https://www.youtube.com/watch?v=GEgFmxxOJno
_______________________________________
Even and Odd Functions:
https://www.youtube.com/watch?v=fKyBOLsqRlo
Transformations of Functions:
https://www.youtube.com/watch?v=Tmdrjs9xufc
Linear Functions:
https://www.youtube.com/watch?v=BtcKotD6Ni8
Functions and Graphs - Problems:
https://www.youtube.com/watch?v=kvU9sOzT2mk
Functions - Test Review:
https://www.youtube.com/watch?v=1xATmTI-YY8
______________________________________
Final Exams and Video Playlists:
https://www.video-tutor.net/
Full-Length Videos and Worksheets:
https://www.patreon.com/MathScienceTutor/collections
In mathematics, and particularly in functional analysis and the calculus of variations, a functional is a function from a vector space into its underlying scalar field, or a set of functions of the real numbers. In other words, it is a function that takes a vector as its input argument, and returns a scalar. Commonly the vector space is a space of functions, thus the functional takes a function for its input argument, then it is sometimes considered a function of a function (a higher-order function). Its use originates in the calculus of variations where one searches for a function that minimizes a certain functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional.
Functional details
Duality
The mapping
is a function, where x0 is an argument of a function f.
At the same time, the mapping of a function to the value of the function at a point
Provided that f is a linear function from a linear vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.
Deepening the foundationsOn the flip side, there’s a burgeoning misconception that creating AI tools doesn’t require a strong foundation in mathematics or computer science.
When they act as the latter, they form wave-like patterns that are described using wave functions —- mathematical descriptions that predict the probabilities of an electron‘s location and other properties at a given time. .
... supremacy, in part because it’s broadly understood that a fully functioning quantum computer could also solve the sort of complex mathematical problems that form the basis of public-key cryptography.
AI works through a combination of algorithms, mathematical models, and computational power. The key to AI’s functionality lies in its ability to learn from data. Here’s a breakdown of the fundamental components that allow AI to function... Finance.
Srinivasa Ramanujan's mathematical talent cannot be defined by just one of his many achievements during his short life ...NationalMathematics Day ... National Mathematics Day 2024 ... National Mathematics Day 2024.
It tests students’ proficiency in Physics, Chemistry, and Mathematics, assessing both their conceptual understanding and problem-solving skills ... Physics, Chemistry, and Mathematics ... removed.Mathematics.
“Until the 19th century, mathematics had concerned itself only with functions that produced differentiable curves,” explained Holly Trochet, then a pure mathematician at the University of St Andrews, in 2009.
Presidential election outcomes are a function of wealth, media support and an algorithm of the swing states. Mathematical programs can predict multiple scenarios for carrying the swing states.
In hidden variable theories (much less popular than the standard interpretation), the wave function is a mathematical description of what we observe, and we are missing something fundamental.
Mathematically, the difference between the real output and the desired output corresponds to an expression called the loss function. It's by applying a mathematical operator known as a gradient to that ...
Courses on simple mathematics and aerobic exercise or strength training were used to enhance brain and body functions, effectively delaying dementia progression and improving mild cognitive impairment, he said.
Golding emphasized that mathematics is ... “In the 21st century, increasing parts of our societal functioning are mathematics-dependent, including the major ‘wicked problems’ of our time.
O. n Dec. 26, 2004, at 7.58.53 a.m ...Scientists came to this conclusion after recalculating tsunami data using Green’s Function, a mathematical method that analyzes how tsunami waves are formed and spread ... From Dec ... .
Another woman, a first-timer, craves more music in her life ... “It crosses the left and right sides because you’re doing creativity and mathematics,” she said. “Drumming is mathematical because rhythm is math ... “It works memory and cognitive function ... ....