Function | Definition, Types, Examples, & Facts | Britannica
Function | Definition, Types, Examples, & Facts | Britannica
Search Britannica
Click here to search
Search Britannica
Click here to search
Login
Subscribe
Subscribe
Home
Games & Quizzes
History & Society
Science & Tech
Biographies
Animals & Nature
Geography & Travel
Arts & Culture
Money
Videos
On This Day
One Good Fact
Dictionary
New Articles
History & Society
Lifestyles & Social Issues
Philosophy & Religion
Politics, Law & Government
World History
Science & Tech
Health & Medicine
Science
Technology
Biographies
Browse Biographies
Animals & Nature
Birds, Reptiles & Other Vertebrates
Bugs, Mollusks & Other Invertebrates
Environment
Fossils & Geologic Time
Mammals
Plants
Geography & Travel
Geography & Travel
Arts & Culture
Entertainment & Pop Culture
Literature
Sports & Recreation
Visual Arts
Companions
Demystified
Image Galleries
Infographics
Lists
Podcasts
Spotlights
Summaries
The Forum
Top Questions
#WTFact
100 Women
Britannica Kids
Saving Earth
Space Next 50
Student Center
Home
Games & Quizzes
History & Society
Science & Tech
Biographies
Animals & Nature
Geography & Travel
Arts & Culture
Money
Videos
function
Table of Contents
function
Table of Contents
IntroductionCommon functionsComplex functionsInverse functionsOther functional expressions
References & Edit History
Related Topics
Images
For Students
function summary
Quizzes
Numbers and Mathematics
Define It: Math Terms
Discover
How Did Helen Keller Fly a Plane?
New Seven Wonders of the World
The Seven Sacraments of the Roman Catholic church
Ten Days That Vanished: The Switch to the Gregorian Calendar
Celebrating Ramadan
Titanosaurs: 8 of the World's Biggest Dinosaurs
Have Any U.S. Presidents Decided Not to Run For a Second Term?
Home
Science
Mathematics
Science & Tech
function
mathematics
Actions
Cite
verifiedCite
While every effort has been made to follow citation style rules, there may be some discrepancies.
Please refer to the appropriate style manual or other sources if you have any questions.
Select Citation Style
MLA
APA
Chicago Manual of Style
Copy Citation
Share
Share
Share to social media
URL
https://www.britannica.com/science/function-mathematics
Give Feedback
External Websites
Feedback
Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login).
Feedback Type
Select a type (Required)
Factual Correction
Spelling/Grammar Correction
Link Correction
Additional Information
Other
Your Feedback
Submit Feedback
Thank you for your feedback
Our editors will review what you’ve submitted and determine whether to revise the article.
External Websites
Whitman College - Mathematics - Functions
University of California, Davis - Department of Mathematics - Sets and Functions
Mathematics LibreTexts - Properties of Functions
Khan Academy - What is a function?
Columbia University - Variables, Functions and Equations
Please select which sections you would like to print:
Table Of Contents
Cite
verifiedCite
While every effort has been made to follow citation style rules, there may be some discrepancies.
Please refer to the appropriate style manual or other sources if you have any questions.
Select Citation Style
MLA
APA
Chicago Manual of Style
Copy Citation
Share
Share
Share to social media
URL
https://www.britannica.com/science/function-mathematics
Feedback
External Websites
Feedback
Corrections? Updates? Omissions? Let us know if you have suggestions to improve this article (requires login).
Feedback Type
Select a type (Required)
Factual Correction
Spelling/Grammar Correction
Link Correction
Additional Information
Other
Your Feedback
Submit Feedback
Thank you for your feedback
Our editors will review what you’ve submitted and determine whether to revise the article.
External Websites
Whitman College - Mathematics - Functions
University of California, Davis - Department of Mathematics - Sets and Functions
Mathematics LibreTexts - Properties of Functions
Khan Academy - What is a function?
Columbia University - Variables, Functions and Equations
Written and fact-checked by
The Editors of Encyclopaedia Britannica
Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree. They write new content and verify and edit content received from contributors.
The Editors of Encyclopaedia Britannica
Last Updated:
Mar 3, 2024
•
Article History
Table of Contents
cubic equation
See all media
Category:
Science & Tech
Key People:
Paul Painlevé
Karl Weierstrass
Émile Borel
Edward Charles Titchmarsh
Peter Gustav Lejeune Dirichlet
(Show more)
Related Topics:
special function
root
continuity
Riemann zeta function
recursive function
(Show more)
See all related content →
function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences. The modern definition of function was first given in 1837 by the German mathematician Peter Dirichlet:If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x.This relationship is commonly symbolized as y = f(x)—which is said “f of x”—and y and x are related such that for every x, there is a unique value of y. That is, f(x) can not have more than one value for the same x. To use the language of set theory, a function relates an element x to an element f(x) in another set. The set of values of x is called the domain of the function, and the set of values of f(x) generated by the values in the domain is called the range of the function. In addition to f(x), other abbreviated symbols such as g(x) and P(x) are often used to represent functions of the independent variable x, especially when the nature of the function is unknown or unspecified. Common functions Many widely used mathematical formulas are expressions of known functions. For example, the formula for the area of a circle, A = πr2, gives the dependent variable A (the area) as a function of the independent variable r (the radius). Functions involving more than two variables (called multivariable or multivariate functions) also are common in mathematics, as can be seen in the formula for the area of a triangle, A = bh/2, which defines A as a function of both b (base) and h (height). In these examples, physical constraints force the independent variables to be positive numbers. When the independent variables are also allowed to take on negative values—thus, any real number—the functions are known as real-valued functions.
Britannica Quiz
Numbers and Mathematics
The formula for the area of a circle is an example of a polynomial function. The general form for such functions is P(x) = a0 + a1x + a2x2+⋯+ anxn, where the coefficients (a0, a1, a2,…, an) are given, x can be any real number, and all the powers of x are counting numbers (1, 2, 3,…). (When the powers of x can be any real number, the result is known as an algebraic function.) Polynomial functions have been studied since the earliest times because of their versatility—practically any relationship involving real numbers can be closely approximated by a polynomial function. Polynomial functions are characterized by the highest power of the independent variable. Special names are commonly used for such powers from one to five—linear, quadratic, cubic, quartic, and quintic for the highest powers being 1, 2, 3, 4, and 5, respectively. Polynomial functions may be given geometric representation by means of analytic geometry. The independent variable x is plotted along the x-axis (a horizontal line), and the dependent variable y is plotted along the y-axis (a vertical line). When the graph of a relation between x and y is plotted in the x-y plane, the relation is a function if a vertical line always passes through only one point of the graphed curve; that is, there would be only one point f(x) corresponding to each x, which is the definition of a function. The graph of the function then consists of the points with coordinates (x, y) where y = f(x). For example, the graph of the cubic equation f(x) = x3 − 3x + 2 is shown in the figure.
Get a Britannica Premium subscription and gain access to exclusive content.
Subscribe Now
graphs of some trigonometric functionsNote that each of these functions is periodic. Thus, the sine and cosine functions repeat every 2π, and the tangent and cotangent functions repeat every π.(more)Another common type of function that has been studied since antiquity is the trigonometric functions, such as sin x and cos x, where x is the measure of an angle (see figure). Because of their periodic nature, trigonometric functions are often used to model behaviour that repeats, or “cycles.” The exponential function is a relation of the form y = ax, with the independent variable x ranging over the entire real number line as the exponent of a positive number a. Probably the most important of the exponential functions is y = ex, sometimes written y = exp (x), in which e (2.7182818…) is the base of the natural system of logarithms (ln). By definition x is a logarithm, and there is thus a logarithmic function that is the inverse of the exponential function. Specifically, if y = ex, then x = ln y. Nonalgebraic functions, such as exponential and trigonometric functions, are also known as transcendental functions.
Complex functions point in the complex planeA point in the complex plane. Unlike real numbers, which can be located by a single signed (positive or negative) number along a number line, complex numbers require a plane with two axes, one axis for the real number component and one axis for the imaginary component. Although the complex plane looks like the ordinary two-dimensional plane, where each point is determined by an ordered pair of real numbers (x, y), the point x + iy is a single number.(more)Practical applications of functions whose variables are complex numbers are not so easy to illustrate, but they are nevertheless very extensive. They occur, for example, in electrical engineering and aerodynamics. If the complex variable is represented in the form z = x + iy, where i is the imaginary unit (the square root of −1) and x and y are real variables (see figure), it is possible to split the complex function into real and imaginary parts: f(z) = P(x, y) + iQ(x, y).
Function Definition & Meaning - Merriam-Webster
Function Definition & Meaning - Merriam-Webster
Menu Toggle
Merriam-Webster Logo
Games & Quizzes
Games & Quizzes
Word of the Day
Grammar
Wordplay
Word Finder
Thesaurus
Join MWU
Shop
Books
Merch
Settings
My Words
Recents
Account
Log Out
More
Thesaurus
Join MWU
Shop
Books
Merch
Log In
Username
My Words
Recents
Account
Log Out
Est. 1828
Dictionary
Definition
noun
verb
noun
2
noun
verb
Synonyms
Synonym Chooser
Example Sentences
Word History
Phrases Containing
Entries Near
Cite this EntryCitation
Share
Kids DefinitionKids
Medical DefinitionMedical
More from M-W
Show more
Show more
Citation
Share
Kids
Medical
More from M-W
Save Word
To save this word, you'll need to log in.
Log In
function
1 of 2
noun
func·tion
ˈfəŋ(k)-shən
Synonyms of function
1
: professional or official position : occupation
His job combines the functions of a manager and a worker.
2
: the action for which a person or thing is specially fitted or used or for which a thing exists : purpose
3
: any of a group of related actions contributing to a larger action
especially
: the normal and specific contribution of a bodily part to the economy of a living organism
The function of the heart is to pump blood through the body.
4
: an official or formal ceremony or social gathering
They went to several functions during their college reunion weekend.
5
a
: a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set
b
: a variable (such as a quality, trait, or measurement) that depends on and varies with another
height is a function of age
also
: result
illnesses that are a function of stress
6
: characteristic behavior of a chemical compound due to a particular reactive unit
also
: functional group
7
: a computer subroutine
specifically
: one that performs a calculation with variables (see variable entry 2 sense 1a) provided by a program and supplies the program with a single result
functionless
ˈfəŋ(k)-shən-ləs
adjective
function
2 of 2
verb
functioned; functioning
ˈfəŋ(k)-sh(ə-)niŋ
intransitive verb
1
: to have a function : serve
an attributive noun functions as an adjective
2
: to carry on a function or be in action : operate
Synonyms
Noun
affair
bash
binge
blast
blowout
do
event
fete
fête
get-together
party
reception
shindig
Verb
act
perform
serve
work
See all Synonyms & Antonyms in Thesaurus
Choose the Right Synonym for function
function, office, duty, province mean the acts or operations expected of a person or thing. function implies a definite end or purpose or a particular kind of work.
the function of language is two-fold: to communicate emotion and to give information
—Aldous Huxley
office is typically applied to the function or service associated with a trade or profession or a special relationship to others.
they exercise the offices of the judge, the priest, the counsellor
—W. E. Gladstone
duty applies to a task or responsibility imposed by one's occupation, rank, status, or calling.
it is the judicial duty of the court, to examine the whole case
—R. B. Taney
province applies to a function, office, or duty that naturally or logically falls to one.
I felt it was not my province to inquire
—Anne Brontë
Examples of function in a Sentence
Noun
The function of the heart is to pump blood through the body.
He believes that the true function of art is to tell the truth.
What functions do these programs fulfill?
infants learning to control their bodily functions
The instrument is chiefly used to measure and record heart function.
The design achieves a perfect blend of form and function.
His job combines the functions of a manager and a worker.
Her chief function is to provide expert legal advice.
They went to several functions during their college reunion weekend.
Verb
The new machine functions well.
His bad health has prevented him from being able to function effectively in recent weeks.
Her heart now seems to be functioning normally.
The computer network is not yet fully functioning.
See More
Recent Examples on the WebNoun
The Husqvarna has an air purge function that removes air from the carburetor and fuel system, reducing resistance in the starter cord by up to 40%.
—Rachel Ahrnsen, Better Homes & Gardens, 28 Feb. 2024
Some 44% of logistics and inventory functions reported the use of management software, while 30% rely on spreadsheets.
—Angelica Mari, Forbes, 28 Feb. 2024
By learning from leading peers, members can be the first to know about new trends, identify next moves, and implement successful strategies that bolster HR’s function with market performances.
—Emma Burleigh, Fortune, 27 Feb. 2024
Antitrust experts said that the broad repercussions of the Change attack show why putting one conglomerate at the center of multiple health care functions is inherently risky.
—Brittany Trang , Tara Bannow, STAT, 27 Feb. 2024
Spacious Weekender Bag $110 $33 The best type of bag is one that serves multiple functions in your day-to-day life, and this sleek and spacious canvas tote from Everlane does just that.
—Merrell Readman, Travel + Leisure, 27 Feb. 2024
In addition, reductions will take place in various functions across Sony Interactive Entertainment in the U.K., the company said.
—Michelle Chapman, Quartz, 27 Feb. 2024
Last week, his government announced the closure of its anti-discrimination agency, saying the Ministry of Justice would absorb its functions.
—Tara John, CNN, 27 Feb. 2024
In the Florida case in particular, the justices spent a significant portion of that time trying to zero in on nuanced differences between social media platforms and functions that appeared to create confusion and division about how to move forward.
—Kaia Hubbard, CBS News, 26 Feb. 2024
Verb
More news to know now The last functioning hospital in northern Gaza has shut down.
—USA TODAY, 29 Feb. 2024
As technology advances, solutions like digital twins—which function as virtual replicas of a physical object, person or process for the purpose of simulation and performance optimization—provide new avenues for AI to prepare for and safeguard more accurately against cyber-attacks.
—Melissa Russell, Forbes, 29 Feb. 2024
With most of Gaza's hospitals forced to close, the few that are still partially functioning do not have enough medical supplies, doctors or staff to cope with the number of dead and wounded, particularly in the north, where Israel ordered evacuations.
—Aya Batrawy, NPR, 29 Feb. 2024
News organizations, crucial to a functioning society, are being hollowed out, if not outright dying.
—Oliver Darcy, CNN, 28 Feb. 2024
Instead of functioning as standalone apps with storage and the ability to send push notifications, web apps in the EU will work more like bookmarks.
—Emma Roth, The Verge, 26 Feb. 2024
The country has been without a president for a year, and a functioning government for almost two, and the currency has lost more than 90 percent of its value since 2019.
—Washington Post, 25 Feb. 2024
The idea: Lower emissions by upending the way data centers function.
—Naureen S. Malik, Fortune, 25 Feb. 2024
There are eight Neighborhood Place locations that will also function as overnight warming centers, Greenberg said.
—Leo Bertucci, The Courier-Journal, 23 Feb. 2024
See More
These examples are programmatically compiled from various online sources to illustrate current usage of the word 'function.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.
Word History
Etymology
Noun and Verb
Latin function-, functio performance, from fungi to perform; probably akin to Sanskrit bhuṅkte he enjoys
First Known Use
Noun
1533, in the meaning defined at sense 2 Verb
1856, in the meaning defined at sense 1
Time Traveler
The first known use of function was
in 1533
See more words from the same year
Phrases Containing function
cumulative distribution function
discriminant function
density function
distribution function
function key
exponential function
logarithmic function
linear function
hyperbolic function
function word
step function
sentential function
rational function
propositional function
probability function
probability density function
power function
trigonometric function
composite function
executive function
a function of
Bessel function
circular function
inverse function
See More
Dictionary Entries Near function
Funchal
function
functionaire
See More Nearby Entries
Cite this Entry
Style
MLA
Chicago
APA
Merriam-Webster
“Function.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/function. Accessed 12 Mar. 2024.
Copy Citation
Share
Post the Definition of function to Facebook
Share the Definition of function on Twitter
Kids Definition
function
1 of 2
noun
func·tion
ˈfəŋ(k)-shən
1
: professional job or duties : occupation
2
a
: the particular purpose for which a person or thing is specially fitted or used or for which a thing exists
the function of a knife is cutting
b
: the natural or proper action of a bodily part in a living thing
the function of the heart
3
: a large important ceremony or social affair
4
a
: a mathematical relationship that assigns exactly one element of one set to each element of the same or another set
b
: something (as a quality, trait, or measurement) that is determined by or based on something else
height is a function of age in children
functionless
-ləs
adjective
function
2 of 2
verb
functioned; functioning
-sh(ə-)niŋ
: to serve a certain purpose : work
Medical Definition
function
1 of 2
noun
func·tion
ˈfəŋ(k)-shən
: any of a group of related actions contributing to a larger action
especially
: the normal and specific contribution of a bodily part to the economy of a living organism see vital function
functionless
-ləs
adjective
function
2 of 2
intransitive verb
functioned; functioning
-sh(ə-)niŋ
: to have a function
shivering functions to maintain the heat of the body
More from Merriam-Webster on function
Nglish: Translation of function for Spanish Speakers
Britannica English: Translation of function for Arabic Speakers
Britannica.com: Encyclopedia article about function
Last Updated:
2 Mar 2024
- Updated example sentences
Love words? Need even more definitions?
Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free!
Merriam-Webster unabridged
Can you solve 4 words at once?
Play
Play
Can you solve 4 words at once?
Play
Play
Word of the Day
ulterior
See Definitions and Examples »
Get Word of the Day daily email!
Popular in Grammar & Usage
See All
8 Grammar Terms You Used to Know, But Forgot
Homophones, Homographs, and Homonyms
Your vs. You're: How to Use Them Correctly
Every Letter Is Silent, Sometimes: A-Z List of Examples
More Commonly Mispronounced Words
See All
Popular in Wordplay
See All
'Arsy-Varsy,' and Other Snappy Reduplicatives
The Words of the Week - Mar. 8
10 Scrabble Words Without Any Vowels
12 More Bird Names that Sound Like Insults (and Sometimes Are)
8 Uncommon Words Related to Love
See All
Games & Quizzes
See All
Quordle
Can you solve 4 words at once?
Play
Blossom Word Game
You can make only 12 words. Pick the best ones!
Play
Missing Letter
A crossword with a twist
Play
Spelling Bee Quiz
Can you outdo past winners of the National Spelli...
Take the quiz
Merriam Webster
Learn a new word every day. Delivered to your inbox!
Help
About Us
Advertising Info
Contact Us
Diversity
Privacy Policy
Terms of Use
YouTube
© 2024 Merriam-Webster, Incorporated
What is a Function? Definition, Types and Notation
What is a Function? Definition, Types and Notation
Login Study MaterialsNCERT SolutionsNCERT Solutions For Class 12 NCERT Solutions For Class 12 PhysicsNCERT Solutions For Class 12 ChemistryNCERT Solutions For Class 12 BiologyNCERT Solutions For Class 12 MathsNCERT Solutions Class 12 AccountancyNCERT Solutions Class 12 Business StudiesNCERT Solutions Class 12 EconomicsNCERT Solutions Class 12 Accountancy Part 1NCERT Solutions Class 12 Accountancy Part 2NCERT Solutions Class 12 Micro-EconomicsNCERT Solutions Class 12 CommerceNCERT Solutions Class 12 Macro-EconomicsNCERT Solutions For Class 11 NCERT Solutions For Class 11 PhysicsNCERT Solutions For Class 11 ChemistryNCERT Solutions For Class 11 BiologyNCERT Solutions For Class 11 MathsNCERT Solutions Class 11 AccountancyNCERT Solutions Class 11 Business StudiesNCERT Solutions Class 11 EconomicsNCERT Solutions Class 11 StatisticsNCERT Solutions Class 11 CommerceNCERT Solutions For Class 10 NCERT Solutions for Class 10 Social ScienceNCERT Solutions for Class 10 Maths NCERT Solutions for Class 10 Maths Chapter 1NCERT Solutions for Class 10 Maths Chapter 2NCERT Solutions for Class 10 Maths Chapter 3NCERT Solutions for Class 10 Maths Chapter 4NCERT Solutions for Class 10 Maths Chapter 5NCERT Solutions for Class 10 Maths Chapter 6NCERT Solutions for Class 10 Maths Chapter 7NCERT Solutions for Class 10 Maths Chapter 8NCERT Solutions for Class 10 Maths Chapter 9NCERT Solutions for Class 10 Maths Chapter 10NCERT Solutions for Class 10 Maths Chapter 11NCERT Solutions for Class 10 Maths Chapter 12NCERT Solutions for Class 10 Maths Chapter 13NCERT Solutions for Class 10 Maths Chapter 14NCERT Solutions for Class 10 Maths Chapter 15MoreNCERT Solutions for Class 10 Science NCERT Solutions for Class 10 Science Chapter 1NCERT Solutions for Class 10 Science Chapter 2NCERT Solutions for Class 10 Science Chapter 3NCERT Solutions for Class 10 Science Chapter 4NCERT Solutions for Class 10 Science Chapter 5NCERT Solutions for Class 10 Science Chapter 6NCERT Solutions for Class 10 Science Chapter 7NCERT Solutions for Class 10 Science Chapter 8NCERT Solutions for Class 10 Science Chapter 9NCERT Solutions for Class 10 Science Chapter 10NCERT Solutions for Class 10 Science Chapter 11NCERT Solutions for Class 10 Science Chapter 12NCERT Solutions for Class 10 Science Chapter 13NCERT Solutions for Class 10 Science Chapter 14NCERT Solutions for Class 10 Science Chapter 15NCERT Solutions for Class 10 Science Chapter 16MoreNCERT Solutions For Class 9 NCERT Solutions For Class 9 Social ScienceNCERT Solutions For Class 9 Maths NCERT Solutions For Class 9 Maths Chapter 1NCERT Solutions For Class 9 Maths Chapter 2NCERT Solutions For Class 9 Maths Chapter 3NCERT Solutions For Class 9 Maths Chapter 4NCERT Solutions For Class 9 Maths Chapter 5NCERT Solutions For Class 9 Maths Chapter 6NCERT Solutions For Class 9 Maths Chapter 7NCERT Solutions For Class 9 Maths Chapter 8NCERT Solutions For Class 9 Maths Chapter 9NCERT Solutions For Class 9 Maths Chapter 10NCERT Solutions For Class 9 Maths Chapter 11NCERT Solutions For Class 9 Maths Chapter 12NCERT Solutions For Class 9 Maths Chapter 13NCERT Solutions For Class 9 Maths Chapter 14NCERT Solutions For Class 9 Maths Chapter 15MoreNCERT Solutions For Class 9 Science NCERT Solutions for Class 9 Science Chapter 1NCERT Solutions for Class 9 Science Chapter 2NCERT Solutions for Class 9 Science Chapter 3NCERT Solutions for Class 9 Science Chapter 4NCERT Solutions for Class 9 Science Chapter 5NCERT Solutions for Class 9 Science Chapter 6NCERT Solutions for Class 9 Science Chapter 7NCERT Solutions for Class 9 Science Chapter 8NCERT Solutions for Class 9 Science Chapter 9NCERT Solutions for Class 9 Science Chapter 10NCERT Solutions for Class 9 Science Chapter 11NCERT Solutions for Class 9 Science Chapter 12NCERT Solutions for Class 9 Science Chapter 13NCERT Solutions for Class 9 Science Chapter 14NCERT Solutions for Class 9 Science Chapter 15MoreNCERT Solutions For Class 8 NCERT Solutions For Class 8 MathsNCERT Solutions For Class 8 ScienceNCERT Solutions for Class 8 Social ScienceNCERT Solutions for Class 8 EnglishNCERT Solutions For Class 7 NCERT Solutions For Class 7 MathsNCERT Solutions For Class 7 ScienceNCERT Solutions for Class 7 Social ScienceNCERT Solutions for Class 7 EnglishNCERT Solutions For Class 6 NCERT Solutions For Class 6 MathsNCERT Solutions For Class 6 ScienceNCERT Solutions For Class 6 Social ScienceNCERT Solutions for Class 6 EnglishNCERT Solutions For Class 5 NCERT Solutions For Class 5 MathsNCERT Solutions For Class 5 EVSNCERT Solutions For Class 4 NCERT Solutions For Class 4 MathsNCERT Solutions For Class 4 EVSNCERT Solutions For Class 3 NCERT Solutions For Class 3 MathsNCERT Solutions For Class 3 EVSNCERT Solutions For Class 3 EnglishNCERT Solutions For Class 2 NCERT Solutions For Class 2 MathsNCERT Solutions For Class 2 EnglishNCERT Solutions For Class 1 NCERT Solutions For Class 1 MathsNCERT Solutions For Class 1 EnglishNCERTNCERT SyllabusClassesClass 1 - 3Class 4 - 5Class 6 - 10CBSENCERT Books NCERT Books for Class 5NCERT Books Class 6NCERT Books for Class 7NCERT Books for Class 8NCERT Books for Class 9NCERT Books for Class 10NCERT Books for Class 11NCERT Books for Class 12NCERT Exemplar NCERT Exemplar Class 8NCERT Exemplar Class 9NCERT Exemplar Class 10NCERT Exemplar Class 11NCERT Exemplar Class 12RD Sharma RD Sharma Class 6 SolutionsRD Sharma Class 7 SolutionsRD Sharma Class 8 SolutionsRD Sharma Class 9 SolutionsRD Sharma Class 10 SolutionsRD Sharma Class 11 SolutionsRD Sharma Class 12 SolutionsPHYSICS MechanicsOpticsThermodynamicsElectromagnetismFamous PhysicistsUnit ConversionKirchhoff's LawsFaraday's LawLaws of MotionRefraction of LightMaxwell's EquationElectrostaticsBernoulli's PrincipleProjectile MotionElectric ChargePhysics SymbolsMoreCHEMISTRY Periodic TableStereochemistryOrganic CompoundsInorganic ChemistryQuantum NumbersAtomic Mass of ElementsPeriodic Properties of Elements118 Elements and Their SymbolsBalancing Chemical EquationsSalt AnalysisAcids, Bases, and SaltsBenzeneOrganometallic CompoundsAtomic Number and Mass NumberMoreMATHS Pythagoras TheoremPrime NumbersProbability and StatisticsFractionsSetsTrigonometric FunctionsRelations and FunctionsSequence and SeriesMultiplication TablesDeterminants and MatricesProfit And LossPolynomial EquationsDividing FractionsSocial Science Manufacturing IndustriesSahara DesertAmazon RiverIndian TribesMughal DynastyMonuments of India ListJudiciaryAutocratic MeaningFundamental Rights of IndiaUnion Territories of IndiaCapitals of Seven Sisters of IndiaMost Populated States In IndiaBIOLOGY MicrobiologyEcologyZoologyPhotosynthesisFORMULAS Maths FormulasAlgebra FormulasTrigonometry FormulasGeometry FormulasCBSE Sample Papers CBSE Sample Papers for Class 6CBSE Sample Papers for Class 7CBSE Sample Papers for Class 8CBSE Sample Papers for Class 9CBSE Sample Papers for Class 10CBSE Sample Papers for Class 11CBSE Sample Papers for Class 12CBSE Previous Year Question Paper CBSE Previous Year Question Papers Class 10CBSE Previous Year Question Papers Class 12Lakhmir Singh Solutions Lakhmir Singh Class 9 SolutionsLakhmir Singh Class 10 SolutionsLakhmir Singh Class 8 SolutionsCBSE Notes Class 6 CBSE NotesClass 7 CBSE NotesClass 8 CBSE NotesClass 9 CBSE NotesClass 10 CBSE NotesClass 11 CBSE NotesClass 12 CBSE NotesCBSE Revision Notes CBSE Class 9 Revision NotesCBSE Class 10 Revision NotesCBSE Class 11 Revision NotesCBSE Class 12 Revision NotesCBSE Extra Questions CBSE Class 8 Maths Extra QuestionsCBSE Class 8 Science Extra QuestionsCBSE Class 9 Maths Extra QuestionsCBSE Class 9 Science Extra QuestionsCBSE Class 10 Maths Extra QuestionsCBSE Class 10 Science Extra QuestionsCBSE Class Class 3Class 4Class 5Class 6Class 7Class 8Class 9Class 10Class 11Class 12Textbook SolutionsCalculatorsBasic Calculators Percentage CalculatorLoan CalculatorEmi CalculatorFraction CalculatorAlgebra CalculatorFactoring CalculatorDistance CalculatorPercentage Increase CalculatorSquare Footage CalculatorSquare Root CalculatorPercentage Change CalculatorRatio CalculatorTriangle CalculatorFractions CalculatorArea CalculatorPercentage Difference CalculatorPercentage Off CalculatorScientific Notation CalculatorSimple Interest CalculatorEquation CalculatorLcm CalculatorRounding CalculatorLong Division CalculatorMixed Fraction CalculatorMoreMaths Calculators Compound Interest CalculatorIntegral CalculatorDerivative CalculatorGraphing CalculatorStandard Deviation CalculatorLimit CalculatorBinary CalculatorMatrix CalculatorCp CalculatorDiscount CalculatorAntiderivative CalculatorExponents CalculatorProbability CalculatorSample Size CalculatorSlope CalculatorArea Of A Circle CalculatorCircumference CalculatorCombination CalculatorInverse Matrix CalculatorMod CalculatorPythagorean Theorem CalculatorConfidence Interval CalculatorDouble Integral CalculatorMatrix Multiplication CalculatorInverse Function CalculatorVolume CalculatorVolume Of A Cylinder CalculatorZ Score CalculatorVariance CalculatorMorePhysics Calculators Power CalculatorVoltage Drop CalculatorChemistry Calculators Molarity CalculatorCommerceClass 11 Commerce Syllabus Class 11 Accountancy SyllabusClass 11 Business Studies SyllabusClass 11 Economics SyllabusClass 12 Commerce Syllabus Class 12 Accountancy SyllabusClass 12 Business Studies SyllabusClass 12 Economics SyllabusCommerce Sample Papers Class 11 Commerce Sample PapersClass 12 Commerce Sample PapersTS Grewal Solutions TS Grewal Solutions Class 12 AccountancyTS Grewal Solutions Class 11 AccountancyStatement Of Cash FlowsWhat Is EntrepreneurshipConsumer ProtectionWhat Is A Fixed AssetWhat Is A Balance SheetWhat Is Fiscal DeficitWhat Are Equity SharesDifference Between Selling And MarketingICSEICSE Sample PapersICSE Question PapersML Aggarwal Solutions ML Aggarwal Solutions Class 10 MathsML Aggarwal Solutions Class 9 MathsML Aggarwal Solutions Class 8 MathsML Aggarwal Solutions Class 7 MathsML Aggarwal Solutions Class 6 MathsSelina Solutions Selina Solution for Class 8Selina Solutions for Class 10Selina Solution for Class 9Frank Solutions Frank Solutions for Class 10 MathsFrank Solutions for Class 9 MathsICSE Class ICSE Class 6ICSE Class 7ICSE Class 8ICSE Class 9ICSE Class 10ISC Class 11ISC Class 12IASIAS ExamUPSC ExamCivil Service ExamUPSC SyllabusIAS TopperFree IAS PrepUPSC Results UPSC Prelims ResultUPSC Mains ResultCurrent Affairs List of Current Affairs ArticlesList Of IAS ArticlesPublic Service Commission KPSC KAS ExamUPPSC PCS ExamMPSC ExamRPSC RAS ExamTNPSC Group 1APPSC Group 1BPSC ExamWBPSC ExamMPPSC ExamJPSC ExamGPSC ExamUPSC Question Papers UPSC Prelims 2022 Question PaperUPSC Prelims 2022 Answer KeyIAS Coaching IAS Coaching BangaloreIAS Coaching DelhiIAS Coaching ChennaiIAS Coaching HyderabadIAS Coaching MumbaiBest IAS coaching ChennaiBest IAS coaching BangaloreBest IAS coaching DelhiIAS QuestionsJEEJEE 2024NMTC 2023 Question PapersNMTC 2023 Sub Junior DivisionNMTC 2023 Primary DivisionNMTC 2023 Junior DivisionJEE 2023JEE Test SeriesJEE Advanced 2023JEE Advanced 2023 Question PaperJEE Advanced 2023 Paper AnalysisJEE Advanced 2022 Question PaperJEE Advanced Question PapersJEE Advanced Answer KeysJEE Advanced ResultsJEE Advanced Marks vs RankJEE Advanced Rank PredictorJEE Advanced Paper AnalysisJEE Main 2024JEE Main 2024 Mock TestJEE Main 2023 Question PapersJEE Main Toppers List 2023JEE Main 2023 Paper AnalysisJEE Main 2022 Question PapersJEE Main 2022 Paper AnalysisJEE Main Rank PredictorJEE Main ResultsJEE Sample PaperJEE Question PaperJEE Main Answer KeyBinomial TheoremJEE ArticlesQuadratic EquationJEE QuestionsNEETNEET 2024NEET 2023NEET Registration 2023NEET Admit CardNEET Test SeriesNEET 2023 Question PaperNEET 2023 Answer KeyNEET 2023 Question Paper AnalysisNEET 2022 Question PaperNEET 2022 Answer KeyNEET 2022 Question paper analysisNEET 2022 ResultsNEET Eligibility CriteriaNEET Question PapersNEET Sample PapersNEET PreparationNEET SyllabusNEET QuestionsGATE 2024GATEGATE Previous Year Question PapersGATE Syllabus GATE Syllabus For CSEGATE Syllabus For ECEGATE Syllabus For Civil EngineeringGATE Syllabus For Mechanical EngineeringGATE Aptitude SyllabusGATE Study Material GATE Notes For CSEGATE Difference Between ArticlesGATE FAQsOlympiad ExamsMaths OlympiadScience OlympiadEntrance Exams In IndiaCOMED-K COMED-K SyllabusCOMED-K Previous Year Question PapersCOMED-K Sample PapersKCET KCET SyllabusKCET Question PapersWBJEE WBJEE SyllabusWBJEE Question PapersGUJCET GUJCET SyllabusGUJCET Question PapersKVPY KVPY SyllabusMHT-CET MHT-CET SyllabusNSTSE NSTSE SyllabusNTSE NTSE SyllabusNTSE Question PaperBCECE BCECE SyllabusUPSEE UPSEE SyllabusGCETHPCETKEAMState BoardsGSEB GSEB SyllabusGSEB Question PaperGSEB Sample PaperGSEB BooksMSBSHSE MSBSHSE SyllabusMSBSHSE TextbooksMSBSHSE Sample PapersMSBSHSE Question PapersAP Board APSCERT BooksAP SSC SyllabusAP 1st Year SyllabusAP 2nd Year SyllabusMP Board MP Board SyllabusMP Board Sample PapersMP Board TextbooksAssam Board Assam Board SyllabusAssam Board TextbooksAssam Board Sample PapersBSEB Bihar Board SyllabusBihar Board TextbooksBihar Board Question PapersBihar Board Model PapersBSE Odisha Odisha Board SyllabusOdisha Board Sample PapersPSEB PSEB SyllabusPSEB TextbooksPSEB Question PapersRBSE Rajasthan Board SyllabusRBSE TextbooksRBSE Question PapersHPBOSE HPBOSE SyllabusHPBOSE TextbooksHPBOSE Question PapersJKBOSE JKBOSE SyllabusJKBOSE Sample PapersJKBOSE Exam PatternTN Board TN Board SyllabusTN Board Question PapersTN Board Sample PapersSamacheer Kalvi BooksJAC JAC SyllabusJAC TextbooksJAC Question PapersTelangana Board Telangana Board SyllabusTelangana Board TextbooksTelangana Board Question PapersKSEEB KSEEB SyllabusKSEEB Model Question PapersKBPE KBPE SyllabusKBPE TextbooksKBPE Question PapersUPMSP UP Board SyllabusUP Board BooksUP Board Question PapersWest Bengal Board West Bengal Board SyllabusWest Bengal Board TextBooksWest Bengal Board Question PapersUBSETBSEGoa BoardNBSECGBSEMBSEMeghalaya BoardManipur BoardHaryana BoardNIOSNIOS A Level Syllabus A Level EVS SyllabusA Level Basic Computer Skill SyllabusA Level Maths SyllabusNIOS B Level Syllabus B Level EVS SyllabusB Level Computer Skill SyllabusB Level Maths SyllabusNIOS C Level Syllabus C Level Social Science SyllabusC Level Science SyllabusC Level Maths SyllabusC Level Basic Computer Skills SyllabusC Level English Language SyllabusNIOS Secondary Course Secondary Course Group B SyllabusSecondary Course Group A SyllabusNIOS Senior Secondary Course Senior Secondary Group F SyllabusSenior Secondary Group E SyllabusSenior Secondary Group D SyllabusSenior Secondary Group C SyllabusSenior Secondary Group B SyllabusSenior Secondary Group A SyllabusGovernment ExamsBank Exams SBI ExamsIBPS ExamsRBI ExamsIBPS RRB ExamSSC Exams SSC JESSC GDSSC CPOSSC CHSLSSC CGLRRB Exams RRB JERRB NTPCRRB ALPInsurance Exams LIC ExamsLIC HFLLIC ADOUPSC CAPFList of Government Exams ArticlesKids LearningClass 1Class 2Class 3English LanguageLetter WritingSpeech Topics For KidsSlogans for KidsEnglish Grammar Parts of SpeechEnglish Grammar ExercisesAcademic QuestionsPhysics QuestionsChemistry QuestionsBiology QuestionsMaths QuestionsScience QuestionsGK QuestionsCommerce QuestionsEnglish QuestionsOnline TuitionHome TuitionFull FormsGeneral Full Forms Physics Full FormsChemistry Full FormsBiology Full FormsEducational Full FormsBanking Full FormsTechnology Full FormsCATCAT 2023CAT SyllabusCAT Exam PatternCAT ExamCAT Percentile PredictorFree CAT PrepByju's App Review on CATEducation NewsSupportComplaint ResolutionCustomer CareBYJU'S Answer ScholarshipBST Class 4-10BNAT Class 11-12BNST | IASMock Test | JEE MainMock Test | JEE AdvancedMock Test | NEETBTCBYJU'S Tuition CentreTuition in North ZoneTuition in Madhya Pradesh Tuition in IndoreTuition in RewaTuition in UjjainTuition in BhopalTuition in JabalpurTuition in GwaliorTuition in SagarTuition in Chhattisgarh Tuition in RaipurTuition in BilaspurTuition in BhilaiTuition in DurgTuition in DelhiTuition in Haryana Tuition in GurgaonTuition in FaridabadTuition in KarnalTuition in SonipatTuition in AmbalaTuition in HisarTuition in PanipatTuition in JagadhriTuition in SirsaTuition in PanchkulaTuition in Himachal Pradesh Tuition in ShimlaTuition in Punjab Tuition in chandigarhTuition in LudhianaTuition in JalandharTuition in AmritsarTuition in BathindaTuition in MohaliTuition in KhararTuition in Rajasthan Tuition in JaipurTuition in AjmerTuition in JodhpurTuition in BikanerTuition in UdaipurTuition in AlwarTuition in Sri GanganagarTuition in BharatpurTuition in SikarTuition in BhilwaraTuition in Uttar Pradesh Tuition in KanpurTuition in LucknowTuition in NoidaTuition in GhaziabadTuition in AligarhTuition in AgraTuition in MeerutTuition in MathuraTuition in FaizabadTuition in SaharanpurTuition in BareillyTuition in Gautam Buddha NagarTuition in RampurTuition in JhansiTuition in AllahabadMoreTuition in Uttarakhand Tuition in RudrapurTuition in HaridwarTuition in DehradunTuition in RoorkeeTuition in West ZoneTuition in Gujarat Tuition in AhmedabadTuition in SuratTuition in AnandTuition in VadodaraTuition in VapiTuition in BhavnagarTuition in RajkotTuition in JamnagarTuition in Maharashtra Tuition in MumbaiTuition in Navi MumbaiTuition in ThaneTuition in PuneTuition in NashikTuition in NagpurTuition in SolapurTuition in LaturTuition in AhmednagarTuition in AurangabadTuition in AmravatiTuition in NandedTuition in AkolaTuition in DhuleMoreTuition in East ZoneTuition in Bihar Tuition in PatnaTuition in MuzaffarpurTuition in DarbhangaTuition in GayaTuition in BhagalpurTuition in PurniaTuition in Jharkhand Tuition in JamshedpurTuition in BokaroTuition in RanchiTuition in DhanbadTuition in Odisha Tuition in BhubaneswarTuition in CuttackTuition in West Bengal Tuition in KolkataTuition in KharagpurTuition in BardhamanTuition in SiliguriTuition in AsansolTuition in Assam Tuition in DibrugarhTuition in South ZoneTuition in Andhra Pradesh Tuition in KurnoolTuition in NelloreTuition in RajahmundryTuition in VijayawadaTuition in VisakhapatnamTuition in KadapaTuition in GunturTuition in AnantapurTuition in TirupatiTuition in Karnataka Tuition in BangaloreTuition in BelagaviTuition in GulbargaTuition in DharwadTuition in MandyaTuition in BellaryTuition in HassanTuition in VijayapuraTuition in HubliTuition in ShimogaTuition in DavanagereTuition in MysoreTuition in MangaloreTuition in Kerala Tuition in TrivandrumTuition in CochinTuition in ThrissurTuition in KozhikodeTuition in KannurTuition in KollamTuition in PalakkadTuition in KottayamTuition in MalappuramTuition in Tamil Nadu Tuition in ChennaiTuition in CoimbatoreTuition in TiruchirappalliTuition in SalemTuition in VelloreTuition in ErodeTuition in KrishnagiriTuition in PuducherryTuition in Telangana Tuition in HyderabadTuition in KarimnagarTuition in KhammamTuition Centre Near YouBuy a Course Success Stories Live QuizNEWLogin+91-9243500460MathsAlgebra Arithmetic Calculus Geometry Probability and Statistics Number System Set Theory Trigonometry Permutation and Combination Mensuration Maths ArticleMaths Index PagesClass 10 Maths Index Class 11 Maths Index Class 12 Maths Index Maths Properties Math Theorems Maths SymbolsAlgebra Symbols Geometry Symbols Probability And Statistics Symbols Set Theory Symbols Maths TablesTables 1 to 20 Tables 2 to 30 Tables 1 to 100 Tables 100 to 200 Tables 200 to 300 Tables 300 to 400 Tables 400 to 500 Tables 500 to 600 Tables 600 to 700 Tables 700 to 800 Tables 800 to 900 Tables 900 to 1000 Addition Table Multiplication Tables Square RootSquare Root of 2 Square Root of 3 Square Root of 4 Square Root Table Diff. Between In Maths Differential Equations Trigonometry Formulas Trigonometry LawsLaw of Sine Law of Cosines Law of Tangent Maths FormulasMaths formulas for class 6 Maths formulas for class 7 Maths formulas for class 8 Maths formulas for class 9 Maths formulas for class 10 Maths formulas for class 11 Maths formulas for class 12 Maths SyllabusClass 6 Maths Syllabus Class 7 Maths Syllabus Class 8 Maths Syllabus Class 9 Maths Syllabus Class 10 Maths Syllabus Class 11 Maths Syllabus Class 12 Maths Syllabus Maths Important QuestionsMaths Questions Important Questions For Class 8 Maths Important Questions For Class 9 Maths Important Questions For Class 10 Maths Important Questions For Class 11 Maths Important Questions For Class 12 Maths Maths CalculatorMaths MCQsClass 8 Maths MCQs Class 9 Maths MCQs Class 10 Maths MCQs Class 11 Maths MCQs Class 12 Maths MCQs
MathsMath ArticleWhat Is A Function
What is a Function?
What is a function? A function is defined as a relation between a set of inputs having one output each. In simple words, a function is a relationship between inputs where each input is related to exactly one output. Every function has a domain and codomain or range. A function is generally denoted by f(x) where x is the input. The general representation of a function is y = f(x).
These functions are also classified into various types, which we will discuss here. Check Relations and Functions lesson for more information.
What is a Function in Maths?
A function in maths is a special relationship among the inputs (i.e. the domain) and their outputs (known as the codomain) where each input has exactly one output, and the output can be traced back to its input.
Types of Functions in Maths
An example of a simple function is f(x) = x2. In this function, the function f(x) takes the value of “x” and then squares it. For instance, if x = 3, then f(3) = 9. A few more examples of functions are: f(x) = sin x, f(x) = x2 + 3, f(x) = 1/x, f(x) = 2x + 3, etc.
There are several types of functions in maths. Some important types are:
Injective function or One to one function: When there is mapping for a range for each domain between two sets.
Surjective functions or Onto function: When there is more than one element mapped from domain to range.
Polynomial function: The function which consists of polynomials.
Inverse Functions: The function which can invert another function.
These were a few examples of functions. It should be noted that there are various other functions like into function, algebraic functions, etc.
Learn here all the functions:
One To One Function
Onto Function
Polynomial Functions
Inverse Functions
Even function
What is a Function in Algebra?
A function is an equation for which any x that can be put into the equation will produce exactly one output such as y out of the equation. It is represented as;
y = f(x)
Where x is an independent variable and y is a dependent variable.
For example:
y = 2x + 1
y = 3x – 2
y = 4y
y = 5/x
What is a function on a graph?
A function f(x) can be represented on a graph by knowing the values of x. As we know, y = f(x), so if start putting the values of x we can get the related value for y. Hence, we can plot a graph using x and y values in a coordinate plane. Let us see an example:
Suppose, y = x + 3
Then,
when x = 0, y = 3
when x = -2, y = -2 + 3 = 1
when x = -1, y = -1 + 3 = 2
when x = 1, y = 1 + 3 = 4
when x = 2, y = 2 + 3 = 5
Thus, with the help of these values, we can plot the graph for function x + 3.
Recommended Videos
Relations and Types of Relations
Functions – Introduction, Co-domain and Range
Functions and Types of Functions
Range of Functions – Concepts & Questions
Frequently Asked Questions on FunctionQ1 What is the definition of a function?
A function can be defined as a relation between a set of inputs where each input has exactly one output.
Q2 How are functions represented?
A function is generally represented as f(x).
Q3 What is the domain and codomain of a function?
A domain of a function is the set of inputs for which the function is defined. A codomain of a function is the set of possible output values.
Test your Knowledge on What is a Function
Q5
Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!
Select the correct answer and click on the “Finish” buttonCheck your score and answers at the end of the quiz
Start Quiz
Congrats!
Visit BYJU’S for all Maths related queries and study materials
Your result is as below
0 out of 0 arewrong
0 out of 0 are correct
0 out of 0 are Unattempted
View Quiz Answers and Analysis
X
Login To View Results
Mobile Number*
Send OTP
Did not receive OTP?
Request OTP on
Voice Call
Login To View Results
Name*
Email ID*
Grade*
Grade
1
2
3
4
5
6
7
8
9
10
11
12
City*
View Result
MATHS Related Links
Linear Polynomial
Algebraic Identities With Examples
Cube Root Of 1
Area Of Regular Hexagon
Frequency Polygon Example Problems
Mathematical Induction
Coprime Numbers From 1 To 100
Conic Sections Class 11
Perfect Number
Set Theory Notation
CBSE Sample Papers
CBSE Sample Papers Class 8 Maths
CBSE Sample Papers Class 9 Maths
CBSE Sample Papers Class 10 Maths
CBSE Sample Papers Class 11 Maths
CBSE Sample Papers Class 12 Maths
CBSE Previous Year Question Papers
CBSE Previous Year Question Papers Class 12 Maths
CBSE Previous Year Question Papers Class 10 Maths
ICSE Sample Papers
ICSE Sample Papers Class 8 Maths
ICSE Sample Papers Class 9 Maths
ICSE Sample Papers Class 10 Maths
ISC Sample Papers Class 11 Maths
ISC Sample Papers Class 12 Maths
ICSE Previous Year Question Papers
ICSE Previous Year Question Papers Class 10
ISC Previous Year Question Papers Class 12 Maths
BYJU'S Tuition Centre for Class 4 to 10
Class 10 Tuition Centre
Class 9 Tuition Centre
Class 8 Tuition Centre
Class 7 Tuition Centre
Class 6 Tuition Centre
Class 5 Tuition Centre
Class 4 Tuition Centre
Join BYJU'S Learning Program
Grade/Exam
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 7
Class 8
Class 9
Class 10
Class 11
Class 12
IAS
CAT
Bank Exam
GATE
Submit
COURSES
CBSEICSECATIASJEENEETCommerceJEE MainNCERTJEE AdvancedUPSC Prelims 2022 Question PaperUPSC Prelims 2022 Answer KeyIAS CoachingCBSE Sample PapersCBSE Question Papers
EXAMS
CAT ExamCAT 2023GATE ExamGATE 2024IAS ExamUPSC ExamUPSC SyllabusUPSC 2023Bank ExamGovernment ExamsEducation NewsCLASSES
Kids LearningClass 1st - 3rdClass 4th - 5thClass 6th - 10thClass 11th - 12thBYJU'S Tuition Centre
EXAM PREPARATION
Free CAT PrepFree IAS PrepMathsPhysicsChemistryBiologyJEE 2024JEE Advanced 2023 Question Paper with AnswersJEE Main Mock TestJEE Main 2023 Question Papers with AnswersJEE Main 2022 Question Papers with AnswersJEE Advanced 2022 Question Paper with AnswersNEET 2023 Question PaperNEET 2023 Question Paper AnalysisNEET 2022 Answer KeyRESOURCES
CAT College PredictorWorksheetsBYJU'S AnswerDSSLHome TuitionAll ProductsCalculatorsFormulas
COMPANY
About UsContact UsContact our Financial PartnersInvestorsComplianceCareersBYJU'S in MediaSocial Initiative - Education for AllBYJU'S APPFAQStudents Stories - The Learning TreeSupportFaces of BYJU'S – Life at BYJU'SBlogBYJU'S GiveFOLLOW US
FREE TEXTBOOK SOLUTIONS
NCERT SolutionsNCERT ExemplarNCERT Solutions for Class 6NCERT Solutions for Class 7NCERT Solutions for Class 8NCERT Solutions for Class 9NCERT Solutions for Class 10NCERT Solutions for Class 11NCERT Solutions for Class 11 EnglishNCERT Solutions for Class 12 EnglishNCERT Solutions for Class 12RD Sharma SolutionsRD Sharma Class 10 SolutionsICSE Selina Solutions
STATE BOARDS
MaharashtraGujaratTamil NaduKarnatakaKeralaAndhra PradeshTelanganaUttar PradeshBiharRajasthan
Madhya PradeshWest Bengal
FOLLOW US
Disclaimer
Privacy Policy
Terms of Services
Sitemap
© 2024, BYJU'S. All rights reserved.
widgets-close-button
Share
Share
Share
Call Us
Register with BYJU'S & Download Free PDFs
*
Send OTP
*
*
*
*
*
Grade
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 7
Class 8
Class 9
Class 10
Class 11
Class 12
IAS
CAT
Bank Exam
GATE
Download Now
Register with BYJU'S & Watch Live Videos
*
Send OTP
*
*
*
*
*
Grade
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Class 7
Class 8
Class 9
Class 10
Class 11
Class 12
IAS
CAT
Bank Exam
GATE
Watch Now
FREE
Signup
Play&
Win
FUNCTION | English meaning - Cambridge Dictionary
FUNCTION | English meaning - Cambridge Dictionary
Dictionary
Translate
Grammar
Thesaurus
+Plus
Cambridge Dictionary +Plus
Shop
Cambridge Dictionary +Plus
My profile
+Plus help
Log out
Cambridge Dictionary +Plus
My profile
+Plus help
Log out
Log in
/
Sign up
English (UK)
Search
Search
English
Meaning of function in English
functionnoun uk
Your browser doesn't support HTML5 audio
/ˈfʌŋk.ʃən/ us
Your browser doesn't support HTML5 audio
/ˈfʌŋk.ʃən/
function noun
(PURPOSE)
Add to word list
Add to word list
B2 [ C ] the natural purpose (of something) or the duty (of a person): function of The function of the veins is to carry blood to the heart. I'm not quite sure what my function is within the company.perform the function of A thermostat performs the function of controlling temperature.
More examplesFewer examplesShould the function of children's television be to entertain or to enlighten?Surely the function of the law is to protect everyone's rights.Do the lights on this cassette deck have any useful function or are they just for show?Zoos fulfil an important function in the protection of rare species.He performs several important functions within the organization.
SMART Vocabulary: related words and phrases
Goals and purposes
affordance
ambition
avail
cause
dedicated
deflect
dream
eye
it's the thought that counts idiom
key performance indicator
KPI
moonshot
objective
point
target
target setting
thought
towards
undedicated
undirected
See more results »
function noun
(CEREMONY)
C1 [ C ] an official ceremony or a formal social event, such as a party or a special meal, at which a lot of people are usually present: attend an official function As a mayor, he has a lot of official functions to attend.social function I see her two or three times a year, usually at social functions.
SMART Vocabulary: related words and phrases
Ceremonies
anti-ritualism
cavalcade
ceremonial
ceremonially
ceremony
guard of honour
hara-kiri
initiate
initiatory
installation
investiture
knight
master of ceremonies
pageant
requiem
ribbon cutting
rite of passage
ritual
ritualistic
troop
See more results »
You can also find related words, phrases, and synonyms in the topics:
Celebrations, parties & special occasions
function noun
(WORK)
[ U ] the way in which something works or operates: function of It's a disease that affects the function of the nervous system. Studies suggest that regular intake of the vitamin significantly improves brain function.
SMART Vocabulary: related words and phrases
Performing a function
act
act as something
discharge
function as something/someone
functional
functionalist
high-performance
hyperfunction
man
manned
multifunction
multifunctional
on it idiom
operate
purpose
run
take
undischarged
work
work a treat idiom
See more results »
function noun
(COMPUTER)
[ C ]
computing
specialized a process that a computer or a computer program uses to complete a task: a search/save/sort function
SMART Vocabulary: related words and phrases
Operating computers
admin
administrator
aliasing
always-on
back someone up
exit
keystroke
left-click
live streamer
logout
loop
malicious
surf
tap
untagged
untechnical
untick
user interface
vectorization
zip file
See more results »
function noun
(RESULT)
a function of something formal
something that results from something else, or is the way it is because of something else: His success is a function of his having worked so hard.
See more
function noun
(VALUE)
[ C ]
mathematics
specialized (in mathematics) a quantity whose value depends on another value and changes with that value: function of x is a function of y.
SMART Vocabulary: related words and phrases
Numerical relationships
absolute maximum
absolute minimum
arithmetic progression
cardinality
commensurable
differential
golden ratio
harmonic progression
intersect
inverse function
inverse proportion
iteration
proportion
proportional
proportionately
prorate
prorated
quantile
rank correlation
subset
See more results »
functionverb [ I ] uk
Your browser doesn't support HTML5 audio
/ˈfʌŋk.ʃən/ us
Your browser doesn't support HTML5 audio
/ˈfʌŋk.ʃən/
to work or operate: You'll soon learn how the office functions. The television was functioning normally until yesterday. I'm so tired today, I can barely function. Synonym
work (OPERATE)Compare
malfunction verb formal
More examplesFewer examplesYou can't function properly when you're deprived of sleep.Obviously the school cannot function without teachers.The theatre continued to function throughout the war.The washing machine wasn't functioning properly.I don't know how you functioned under those conditions.
SMART Vocabulary: related words and phrases
Performing a function
act
act as something
discharge
function as something/someone
functional
functionalist
high-performance
hyperfunction
man
manned
multifunction
multifunctional
on it idiom
operate
purpose
run
take
undischarged
work
work a treat idiom
See more results »
You can also find related words, phrases, and synonyms in the topics:
Machines - Functioning
Phrasal verb
function as something/someone
(Definition of function from the Cambridge Advanced Learner's Dictionary & Thesaurus © Cambridge University Press)
function | American Dictionary
functionnoun us
Your browser doesn't support HTML5 audio
/ˈfʌŋk·ʃən/
function noun
(PURPOSE)
Add to word list
Add to word list
[ C/U ] a purpose or duty, or the way something or someone works: [ U ] The function of the veins is to carry blood to the heart. [ C ] One of your functions as receptionist is to answer the phone.
function noun
(MATHEMATICAL RELATIONSHIP)
[ C ]
mathematics a relationship between two sets in which each part of the first set is connected with just one member of the second set in number pairs
function noun
(CEREMONY)
[ C ] a social event or official ceremony: Morse went to the White House for a ceremonial function.
functionverb [ I ] us
Your browser doesn't support HTML5 audio
/ˈfʌŋk·ʃən/
function verb [I]
(PERFORM PURPOSE)
to perform the purpose of a particular thing, or to perform the duties of a particular person: She quickly learned how the office functions. I’m so tired today, I can barely function. Our spare bedroom also functions as a study (= is also used for that purpose).
(Definition of function from the Cambridge Academic Content Dictionary © Cambridge University Press)
function | Business English
functionnoun uk
Your browser doesn't support HTML5 audio
/ˈfʌŋkʃən/ us
Your browser doesn't support HTML5 audio
Add to word list
Add to word list
[ C or U ] the purpose that something has: The perfect marriage between product design and function account for the company’s success in the computer market. A fundamental function of the Bank of England is the responsibility of carrying out monetary policy operations.
[ C or U ] a job or task that someone or something does: My function is to help coordinate efficient communication between the departments. carry out/perform a function
[ C ]
HR a particular area of responsibility of a company: a sales/marketing/business functioncore/corporate functions Of all the core functions of most companies, innovation has arguably the most competitive value.
[ C ]
IT a process which a computer or a software program uses to complete a task: a search/save/sort function The website benefits from a highly-efficient search function.
[ C ] an official occasion or event attended by many people: a charity/social/work function
be a function of sth
if something is a function of something else, it is produced as the result of that thing or process: The increased availability of part-time jobs is a function of the structural changes in the labour market.
See also
consumption function
functionverb [ I ] uk
Your browser doesn't support HTML5 audio
/ˈfʌŋkʃən/ us
Your browser doesn't support HTML5 audio
to work correctly and as expected: function effectively/properly/well Flights were delayed because the airport computer system was not functioning.
Phrasal verb
function as sth
(Definition of function from the Cambridge Business English Dictionary © Cambridge University Press)
Examples of function
function
All plots are a function of local wavenumber.
From the Cambridge English Corpus
The functions $n2 and f 3 are also of order 114.
From the Cambridge English Corpus
The constant function 0 is a compensation function.
From the Cambridge English Corpus
The commonly used weight functions in one dimension are as follows.
From the Cambridge English Corpus
Each segment is chosen such that it can be given as a function of the independent variables.
From the Cambridge English Corpus
Reasoning about function and its applications to engineering.
From the Cambridge English Corpus
Again, the accuracy with which this may be done will vary as a function of rate of target occurrence and sampling density.
From the Cambridge English Corpus
If you are not clear on what a function should do, you will find it difficult to write it!
From the Cambridge English Corpus
The above reasoning may not apply if the primitive application occurs within the body of a function.
From the Cambridge English Corpus
The filter function is a (standard) higher-order auxiliary.
From the Cambridge English Corpus
Higher order functions are also a common concept in functional programming.
From the Cambridge English Corpus
What if e was a partial application of a function?
From the Cambridge English Corpus
When the finite domain of a function is the product of several finite sets, we realise the function in its curried form.
From the Cambridge English Corpus
How to use polymorphism when passing functions as arguments (section 9).
From the Cambridge English Corpus
The number () represents the number of basis functions used.
From the Cambridge English Corpus
See all examples of function
These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
Collocations with function
function
These are words often used in combination with function.Click on a collocation to see more examples of it.
adaptive functionFirst, it is certain that if dreaming has an adaptive function, it should have enhanced biological fitness in the environment of our ancestors.
From the Cambridge English Corpus
administrative functionThis is an administrative function.
From the Hansard archive
Example from the Hansard archive. Contains Parliamentary information licensed under the Open Parliament Licence v3.0
autonomic functionThis study is the first to perform detailed electrophysiologic evaluation of autonomic function in children undergoing radiofrequency catheter ablation.
From the Cambridge English Corpus
These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
See all collocations with function
What is the pronunciation of function?
B2,C1
Translations of function
in Chinese (Traditional)
目的, 功能，用途, 職責…
See more
in Chinese (Simplified)
目的, 功能，用途, 职责…
See more
in Spanish
función, acto, recepción…
See more
in Portuguese
função, função [feminine], cerimônia [feminine]…
See more
in more languages
in Marathi
in Japanese
in Turkish
in French
in Catalan
in Dutch
in Tamil
in Hindi
in Gujarati
in Danish
in Swedish
in Malay
in German
in Norwegian
in Urdu
in Ukrainian
in Russian
in Telugu
in Arabic
in Bengali
in Czech
in Indonesian
in Thai
in Vietnamese
in Polish
in Korean
in Italian
कार्य, काम, कार्यक्रम…
See more
機能, 機能（きのう）, 行事（ぎょうじ）…
See more
işlev, büyük resmi davet/tören/parti, çalışmak…
See more
fonction [feminine], réunion [feminine], fonctionner…
See more
funció…
See more
functie, werken…
See more
இயற்கை நோக்கம் (ஏதோ ஒன்றின்) அல்லது (ஒரு நபரின்) கடமை, ஒரு உத்தியோகபூர்வ விழா அல்லது ஒரு விருந்து அல்லது ஒரு சிறப்பு உணவு போன்ற ஒரு முறையான சமூக நிகழ்வு, இதில் பொதுவாக நிறைய மக்கள் கலந்து கொள்வார்கள்…
See more
(किसी वस्तु या व्यक्ति का) स्वाभाविक उद्देश्य या कर्तव्य या कार्य, (औपचारिक) समारोह, उत्सव…
See more
કાર્ય, કર્તવ્ય, કોઈ વસ્તુ અથવા વ્યક્તિ નો કુદરતી હેતુ અથવા ફરજ અથવા કાર્ય…
See more
funktion, opgave, fungere…
See more
funktion, fungera…
See more
fungsi, berfungsi…
See more
die Funktion, funktionieren…
See more
funksjon [masculine], tilstelning [masculine], fungere…
See more
فرض منصبی, کام, تقریب…
See more
функція, призначення, функціонувати…
See more
функция, назначение, прием…
See more
పని/ దేనిదైనా సహజ ప్రయోజనం లేదా ఒక వ్యక్తి చేసే విధి, అధికారిక వేడుక లేదా పార్టీ లేదా ప్రత్యేక భోజనం వంటి అధికారిక సామాజిక కార్యక్రమం, ఈ సమయంలో చాలా మంది సాధారణంగా ఉంటారు…
See more
عَمَل, وَظيفة…
See more
উদ্দেশ্য বা কর্তব্য, ভূমিকা, অনুষ্ঠান…
See more
funkce, fungovat…
See more
fungsi, bekerja…
See more
หน้าที่, ทำงานได้…
See more
chức năng, hoạt động, chạy…
See more
funkcja, rola, uroczystość…
See more
기능…
See more
funzione, compito, funzionare…
See more
Need a translator?
Get a quick, free translation!
Translator tool
Browse
fun run
fun sponge
fun-loving
Funafuti
function
function as something/someone
function key
function room
function word
More meanings of function
All
function key
function room
function word
state function
wave function
bodily function
bounded function
See all meanings
Phrasal Verbs
function as sth
function as something/someone
See all phrasal verb meanings
Idioms and phrases
a function of something phrase
Word of the Day
response
UK
Your browser doesn't support HTML5 audio
/rɪˈspɒns/
US
Your browser doesn't support HTML5 audio
/rɪˈspɑːns/
an answer or reaction
About this
Blog
Forget doing it or forget to do it? Avoiding common mistakes with verb patterns (2)
March 06, 2024
Read More
New Words
inverse vaccine
March 11, 2024
More new words
has been added to list
To top
Contents
EnglishAmericanBusinessExamplesCollocationsTranslations
© Cambridge University Press & Assessment 2024
Learn
Learn
Learn
New Words
Help
In Print
Word of the Year 2021
Word of the Year 2022
Word of the Year 2023
Develop
Develop
Develop
Dictionary API
Double-Click Lookup
Search Widgets
License Data
About
About
About
Accessibility
Cambridge English
Cambridge University Press & Assessment
Consent Management
Cookies and Privacy
Corpus
Terms of Use
© Cambridge University Press & Assessment 2024
Cambridge Dictionary +Plus
My profile
+Plus help
Log out
Dictionary
Definitions
Clear explanations of natural written and spoken English
English
Learner’s Dictionary
Essential British English
Essential American English
Translations
Click on the arrows to change the translation direction.
Bilingual Dictionaries
English–Chinese (Simplified)
Chinese (Simplified)–English
English–Chinese (Traditional)
Chinese (Traditional)–English
English–Dutch
Dutch–English
English–French
French–English
English–German
German–English
English–Indonesian
Indonesian–English
English–Italian
Italian–English
English–Japanese
Japanese–English
English–Norwegian
Norwegian–English
English–Polish
Polish–English
English–Portuguese
Portuguese–English
English–Spanish
Spanish–English
English–Swedish
Swedish–English
Semi-bilingual Dictionaries
English–Arabic
English–Bengali
English–Catalan
English–Czech
English–Danish
English–Gujarati
English–Hindi
English–Korean
English–Malay
English–Marathi
English–Russian
English–Tamil
English–Telugu
English–Thai
English–Turkish
English–Ukrainian
English–Urdu
English–Vietnamese
Translate
Grammar
Thesaurus
Pronunciation
Cambridge Dictionary +Plus
Shop
Cambridge Dictionary +Plus
My profile
+Plus help
Log out
Log in /
Sign up
English (UK)
Change
English (UK)
English (US)
Español
Русский
Português
Deutsch
Français
Italiano
中文 (简体)
正體中文 (繁體)
Polski
한국어
Türkçe
日本語
Tiếng Việt
Nederlands
Svenska
Dansk
Norsk
हिंदी
বাঙ্গালি
मराठी
ગુજરાતી
தமிழ்
తెలుగు
Українська
Follow us
Choose a dictionary
Recent and Recommended
Definitions
Clear explanations of natural written and spoken English
English
Learner’s Dictionary
Essential British English
Essential American English
Grammar and thesaurus
Usage explanations of natural written and spoken English
Grammar
Thesaurus
Pronunciation
British and American pronunciations with audio
English Pronunciation
Translation
Click on the arrows to change the translation direction.
Bilingual Dictionaries
English–Chinese (Simplified)
Chinese (Simplified)–English
English–Chinese (Traditional)
Chinese (Traditional)–English
English–Dutch
Dutch–English
English–French
French–English
English–German
German–English
English–Indonesian
Indonesian–English
English–Italian
Italian–English
English–Japanese
Japanese–English
English–Norwegian
Norwegian–English
English–Polish
Polish–English
English–Portuguese
Portuguese–English
English–Spanish
Spanish–English
English–Swedish
Swedish–English
Semi-bilingual Dictionaries
English–Arabic
English–Bengali
English–Catalan
English–Czech
English–Danish
English–Gujarati
English–Hindi
English–Korean
English–Malay
English–Marathi
English–Russian
English–Tamil
English–Telugu
English–Thai
English–Turkish
English–Ukrainian
English–Urdu
English–Vietnamese
Dictionary +Plus
Word Lists
Choose your language
English (UK)
English (US)
Español
Русский
Português
Deutsch
Français
Italiano
中文 (简体)
正體中文 (繁體)
Polski
한국어
Türkçe
日本語
Tiếng Việt
Nederlands
Svenska
Dansk
Norsk
हिंदी
বাঙ্গালি
मराठी
ગુજરાતી
தமிழ்
తెలుగు
Українська
Contents
English
Noun
function (PURPOSE)
function (CEREMONY)
function (WORK)
function (COMPUTER)
function (RESULT)
a function of something
function (VALUE)
Verb
American
Noun
function (PURPOSE)
function (MATHEMATICAL RELATIONSHIP)
function (CEREMONY)
Verb
function (PERFORM PURPOSE)
Business
Noun
function
be a function of sth
Verb
Examples
Collocations
Translations
Grammar
All translations
My word lists
Add function to one of your lists below, or create a new one.
More
Go to your word lists
Tell us about this example sentence:
The word in the example sentence does not match the entry word.
The sentence contains offensive content.
Cancel
Submit
The word in the example sentence does not match the entry word.
The sentence contains offensive content.
Cancel
Submit
What is a Function
What is a Function
What is a Function?
A function relates an input to an output.
It is like a machine that has an input and an output.
And the output is related somehow to the input.
f(x)
"f(x) = ... " is the classic way of writing a function.
And there are other ways, as you will see!
Input, Relationship, Output
We will see many ways to think about functions, but there are always three main parts:
The input
The relationship
The output
Example: "Multiply by 2" is a very simple function.
Here are the three parts:
Input
Relationship
Output
0
× 2
0
1
× 2
2
7
× 2
14
10
× 2
20
...
...
...
For an input of 50, what is the output?
Some Examples of Functions
x2 (squaring) is a function
x3+1 is also a function
Sine, Cosine and Tangent are functions used in trigonometry
and there are lots more!
But we are not going to look at specific functions ...
... instead we will look at the general idea of a function.
Names
First, it is useful to give a function a name.
The most common name is "f", but we can have other names like "g" ... or even "marmalade" if we want.
But let's use "f":
We say "f of x equals x squared"
what goes into the function is put inside parentheses () after the name of the function:
So f(x) shows us the function is called "f", and "x" goes in
And we usually see what a function does with the input:
f(x) = x2 shows us that function "f" takes "x" and squares it.
Example: with f(x) = x2:
an input of 4
becomes an output of 16.
In fact we can write f(4) = 16.
The "x" is Just a Place-Holder!
Don't get too concerned about "x", it is just there to show us where the input goes and what happens to it.
It could be anything!
So this function:
f(x) = 1 - x + x2
Is the same function as:
f(q) = 1 - q + q2
h(A) = 1 - A + A2
w(θ) = 1 - θ + θ2
The variable (x, q, A, etc) is just there so we know where to put the values:
f(2) = 1 - 2 + 22 = 3
Sometimes There is No Function Name
Sometimes a function has no name, and we see something like:
y = x2
But there is still:
an input (x)
a relationship (squaring)
and an output (y)
Relating
At the top we said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what we put into it!
A function relates an input to an output.
Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16
Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h:
h(age) = age × 20
So, if the age is 10 years, the height is:
h(10) = 10 × 20 = 200 cm
Here are some example values:
age
h(age) = age × 20
0
0
1
20
3.2
64
15
300
...
...
What Types of Things Do Functions Process?
"Numbers" seems an obvious answer, but ...
... which numbers?
For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero.
... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.
So we need something more powerful, and that is where sets come in:
A set is a collection of things.
Here are some examples:
Set of even numbers: {..., -4, -2, 0, 2, 4, ...}
Set of clothes: {"hat","shirt",...}
Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
Positive multiples of 3 that are less than 10: {3, 6, 9}
Each individual thing in the set (such as "4" or "hat") is called a member, or element.
So, a function takes elements of a set, and gives back elements of a set.
A Function is Special
But a function has special rules:
It must work for every possible input value
And it has only one relationship for each input value
This can be said in one definition:
Formal Definition of a Function
A function relates each element of a set
with exactly one element of another
set
(possibly the same set).
The Two Important Things!
1.
"...each element..." means that every element in X is related to some element in Y.
We say that the function covers X (relates every element of it).
(But some elements of Y might not be related to at all, which is fine.)
2.
"...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input.
So "f(2) = 7 or 9" is not right!
"One-to-many" is not allowed, but "many-to-one" is allowed:
(one-to-many)
(many-to-one)
This is NOT OK in a function
But this is OK in a function
When a relationship does not follow those two rules then it is not a function ... it is still a relationship, just not a function.
Example: The relationship x → x2
Could also be written as a table:
X: x
Y: x2
3
9
1
1
0
0
4
16
-4
16
...
...
It is a function, because:
Every element in X is related to Y
No element in X has two or more relationships
So it follows the rules.
(Notice how both 4 and -4 relate to 16, which is allowed.)
Example: This relationship is not a function:
It is a relationship, but it is not a function, for these reasons:
Value "3" in X has no relation in Y
Value "4" in X has no relation in Y
Value "5" is related to more than one value in Y
(But the fact that "6" in Y has no relationship does not matter)
Vertical Line Test
On a graph, the idea of single valued means that no vertical line ever crosses more than one value.
If it crosses more than once it is still a valid curve, but is not a function.
Some types of functions have stricter rules, to find out more you can read Injective, Surjective and Bijective
Infinitely Many
My examples have just a few values, but functions usually work on sets with infinitely many elements.
Example: y = x3
The input set "X" is all Real Numbers
The output set "Y" is also all the Real Numbers
We can't show ALL the values, so here are just a few examples:
X: x
Y: x3
-2
-8
-0.1
-0.001
0
0
1.1
1.331
3
27
and so on...
and so on...
Domain, Codomain and Range
In our examples above
the set "X" is called the Domain,
the set "Y" is called the Codomain, and
the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.
We have a special page on Domain, Range and Codomain if you want to know more.
So Many Names!
Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.
Here are some common terms you should get familiar with:
Example: z = 2u3:
"u" could be called the "independent variable"
"z" could be called the "dependent variable" (it depends on the value of u)
Example: f(4) = 16:
"4" could be called the "argument"
"16" could be called the "value of the function"
Example: h(year) = 20 × year:
h() is the function
"year" could be called the "argument", or the "variable"
For software "year" is called the parameter, and the value given to the parameter is called the variable!
We often call a function "f(x)" when in fact the function is really "f"
Ordered Pairs
And here is another way to think about functions:
Write the input and output of a function as an "ordered pair", such as (4,16).
They are called ordered pairs because the input always comes first, and the output second:
(input, output)
So it looks like this:
( x, f(x) )
Example:
(4,16) means that the function takes in "4" and gives out "16"
Set of Ordered Pairs
A function can then be defined as a set of ordered pairs:
Example: {(2,4), (3,5), (7,3)} is a function that says
"2 is related to 4", "3 is related to 5" and "7 is related 3".
Also, notice that:
the domain is {2,3,7} (the input values)
and the range is {4,5,3} (the output values)
But the function has to be single valued, so we also say
"if it contains (a, b) and (a, c), then b must equal c"
Which is just a way of saying that an input of "a" cannot produce two different results.
Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5.
In other words it is not a function because it is not single valued
A Benefit of Ordered Pairs
We can graph them...
... because they are also coordinates!
So a set of coordinates is also a function (if they follow
the rules above, that is)
A Function Can be in Pieces
We can create functions that behave differently depending on the input value
Example: A function with two pieces:
when x is less than 0, it gives 5,
when x is 0 or more it gives x2
Here are some example values:
x
y
-3
5
-1
5
0
0
2
4
4
16
...
...
Read more at Piecewise Functions.
Explicit vs Implicit
One last topic: the terms "explicit" and "implicit".
Explicit is when the function shows us how to go directly from x to y, such as:
y = x3 − 3
When we know x, we can find y
That is the classic y = f(x) style that we often work with.
Implicit is when it is not given directly such as:
x2 − 3xy + y3 = 0
When we know x, how do we find y?
It may be hard (or impossible!) to go directly from x to y.
"Implicit" comes from "implied", in other words shown indirectly.
Graphing
The Function Grapher can only handle explicit functions,
The Equation Grapher can handle both types (but takes a little longer, and sometimes gets it wrong).
Conclusion
a function relates inputs to outputs
a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).
all the outputs (the actual values related to) are together called the range
a function is a special type of relation where:
every element in the domain is included, and
any input produces only one output (not this or that)
an input and its matching output are together called an ordered pair
so a function can also be seen as a set of ordered pairs
5571, 5572, 535, 5207, 5301, 1173, 7281, 533, 8414, 8430
Injective, Surjective and Bijective
Domain, Range and Codomain
Introduction to Sets
Sets Index
Copyright © 2022 Rod Pierce