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Function | Definition, Types, Examples, & Facts | Britannica
Function | Definition, Types, Examples, & Facts | Britannica
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function
Table of Contents
IntroductionCommon functionsComplex functionsInverse functionsOther functional expressions
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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.
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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.
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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
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Est. 1828
Dictionary
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verb
noun
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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
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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
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Merriam-Webster
“Function.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/function. Accessed 12 Mar. 2024.
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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
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What is a Function? Definition, Types and Notation
What is a Function? Definition, Types and Notation
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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.
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FUNCTION | English meaning - Cambridge Dictionary
FUNCTION | English meaning - Cambridge Dictionary
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English (UK)
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English
Meaning of function in English
functionnoun uk
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/ˈfʌŋk.ʃən/ us
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/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
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/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
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/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
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/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
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/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
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/ˈfʌŋkʃən/ us
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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…
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in Chinese (Simplified)
目的, 功能，用途, 职责…
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in Spanish
función, acto, recepción…
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in Portuguese
função, função [feminine], cerimônia [feminine]…
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कार्य, काम, कार्यक्रम…
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機能, 機能（きのう）, 行事（ぎょうじ）…
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işlev, büyük resmi davet/tören/parti, çalışmak…
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fonction [feminine], réunion [feminine], fonctionner…
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funció…
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functie, werken…
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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…
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(किसी वस्तु या व्यक्ति का) स्वाभाविक उद्देश्य या कर्तव्य या कार्य, (औपचारिक) समारोह, उत्सव…
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કાર્ય, કર્તવ્ય, કોઈ વસ્તુ અથવા વ્યક્તિ નો કુદરતી હેતુ અથવા ફરજ અથવા કાર્ય…
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funktion, opgave, fungere…
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funktion, fungera…
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fungsi, berfungsi…
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die Funktion, funktionieren…
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funksjon [masculine], tilstelning [masculine], fungere…
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فرض منصبی, کام, تقریب…
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функція, призначення, функціонувати…
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функция, назначение, прием…
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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…
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عَمَل, وَظيفة…
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উদ্দেশ্য বা কর্তব্য, ভূমিকা, অনুষ্ঠান…
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funkce, fungovat…
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fungsi, bekerja…
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หน้าที่, ทำงานได้…
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chức năng, hoạt động, chạy…
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funkcja, rola, uroczystość…
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기능…
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funzione, compito, funzionare…
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fun run
fun sponge
fun-loving
Funafuti
function
function as something/someone
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function room
function word
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function key
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Phrasal Verbs
function as sth
function as something/someone
See all phrasal verb meanings
Idioms and phrases
a function of something phrase
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response
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/rɪˈspɒns/
US
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/rɪˈspɑːns/
an answer or reaction
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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
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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