Revised 2015-12-24 Introduction to this website website TOC website index blog Back to head of functions chapter
Functions are called by many other names in the literature. This section mentions some of the most common words used to mean "function". Authors may not explain the terminology they use. (See no standardization.)
A function is very commonly referred to as a map or mapping. In some cases, an author will impose conditions on a function to be called a "map". For example, some require that a mapping be a continuous function. Another usage is that a map have a specified codomain. Other uses of the word are described in Wikipedia.
The name "map" is suggested by common metaphors for functions.
The word transformation is commonly used when the domain and the codomain are the same (as in a transformation group ), but that is not the case when it is used in the phrase linear transformation. See also Images and metaphors for functions.
A function between vector spaces may be called an operator. Most functions called "operator" are linear, and the word seems to be most commonly used when the domain is a function space.
Taking the derivative is a linear operator on some suitable vector space of differentiable functions. The article in Wikipedia gives an extensive list of examples.
The word functional is used as a noun to denote some special class of functions. The most common use seems to be to denote a linear function whose domain consists of some vector space and whose values are elements of the field. But the word is used in other senses, as well. See the Wikipedia entry.
In mathematical English, the word "functional" is a noun. In ordinary English, it is almost always an adjective whose meaning has little to do with the mathematical meaning.
A function of the form $f:S\times S\to S$ may be called a binary operation on $S$. ($S\times S$ denotes the cartesian product). The main point to notice is that it takes pairs of elements of $S$ to the same set $S$. See Examples of functions.
In the 1960's some mathematicians were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don't think any mathematician would react this way today.
A binary operation is a special case of n-ary operation for any natural number $n$, which is a function of the form $f:S^n\to S$. A $1$-ary (unary) operation on $S$ is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a $0$-ary operation on $S$ is a constant.
It is useful at times to consider multisorted algebra, where a binary operation can be a function $f:S_1\times S_2\to S_3$ where the $S_i$ are possibly different sets.
The notation $f:S\to T$ means $f$ is a function with domain $S$ and codomain $T$. The is arrow notation, sometimes called straight arrow notation to distinguish it from barred arrow notation.
The expression "$f :A\to B$" can be used as a name or as an independent sentence.
A mathematician might write, "Let $h:\mathbb{Z}\to\mathbb{N}$ be defined by $h(n)=n^2$." This means that the domain of $h$ is $\mathbb{Z}$, the codomain is $\mathbb{N}$ and the value at $n$ is $n^2$. In this sentence the phrase $h:\mathbb{Z}\to\mathbb{N}$ would be pronounced "$h$ from $\mathbb{Z}$ to $\mathbb{N}$".
Note that if someone writes, "Let $h:\mathbb{Z}\to\mathbb{N}$ be a function", they have not defined the function. To define it you must also give some way of determining the value at each input; knowing the domain and codomain are not enough.
When a math text "defines a function" it may or may not tell you what the domain or the codomain are. Sometimes, the context makes the domain or codomain clear and other times it doesn't really matter what it is. More about this in Functions: Specification and Definition.
A function defined on a finite set may be given by a table; for example, the finite function defined in the examples section. The table determines the domain of the function (the domain is the set of all first coordinates of the ordered pairs in the table), but not the codomain.
That example also shows a way of defining a finite function using arrows (its cograph}
Warning: the word "cograph" means something else to graph theorists.
A function may be defined by giving an algebraic expression (its formula) that determines its value. An example is the function $g$ defined by $g(x)={{x}^{2}}+2x-4$.
You may define a function using an algorithm, which to start with you can think of as a computer program that calculates the function's value at every input.
One example most everyone sees in high school or college is Newton's method for finding the root of a polynomial. You can find examples on the web of programs implementing Newton's method in C and Mathematica (and dozens of other languages).
If you think more about this idea, you run into subtleties:
The circumference function $C(r)$ could be defined using a geometric definition this way: "$C(r)$ is the circumference of a circle with radius $r$". Of course it can be given by a formula, too: $C(r)=2\pi r$.
Be clear that the geometric definition is just as good a definition as the formula is.. It defines the function as exactly as the formula $C(r)=2\pi r$ does, although of course if you don't know the formula, you have to do some reasoning to figure out what $f(3)$ is (for example).
Another naming technique is barred arrow notation. If $E$ is some mathematical expression that has a definite value for each $x$ in the domain, then you can refer to the function $ x\mapsto E $ without having to give it a name. Barred arrow notation may not be familiar to you, but it is becoming more common. Like the defining expression, it allows you to refer to a function without giving it a name, so it is a form of anonymous notation.
"The function $ x\mapsto {{x}^{2}}+2$ is positive for all real numbers $x$." Here $E$ is the expression ${{x}^{2}}+2 $.
I could also have written "The function $x\mapsto {{x}^{2}}+2:\mathbb{R}\to \mathbb{R}$ is always positive" using the barred arrow notation together with the straight arrow notation.
The straight arrow goes from domain to codomain. The barred arrow goes from element of the domain to element of the codomain. |
Since "$ x\mapsto {{x}^{2}}+2$" is the name of a function, you can use it to show a value at the input, for example, $(x\mapsto {{x}^{2}}+2)(3)=11$. This usage is not common, but it ought to be!
Using the barred arrow clears up ambiguity when the defining expression has parameters in it.
Let $(x\mapsto {{x}^{2}}+yx+z):\mathbb{R}\to \mathbb{R}$. This notation tells you that $x$ is the function variable and $y$ and $z$ are parameters.
Of course, if you had written, "Consider the function ${{x}^{2}}+ax+b$" the experienced reader will assume you mean that $x$ is the variable and $a$ and $b$ are parameters, because of the convention that $x$ is a variable and $a$ and $b$ are parameters. The barred arrow notation does not depend on knowledge of conventions. Even so, the kind math writer would use $a$ and $b$ in the barred arrow version above: "Let $(x\mapsto {{x}^{2}}+ax+b):\mathbb{R}\to \mathbb{R}$. "
A variant of barred arrow notation is to define functions on finite sets element by element. For example the finite function $F$ could be defined by: $1\mapsto 3,\,\,\,2\mapsto 3,\,\,\,3\mapsto 2,\,\,\,6\mapsto 1$.
Functions may have names, for example "sine" or "the exponential function". The name in English and the symbol for the function in the symbolic language may be different; for example, "sine" is the name of the sine function, but in the symbolic language it is called "sin".
A function may be named by a letter of some alphabet, for temporary use in that particular section of text.
Symbolic expressions that are not functions may also be given names. The expression "$E$" mentioned under barred arrow notation is an example. Using letters for naming functions and expressions are examples of local identifiers.
By convention in particular subfields of math, some letters are assumed by default to be the names of certain commonly used functions in that field.
An article about complex functions might refer to "the $\Gamma$ function" without defining it. The author expects that the reader will know that it refers to a certain well-known function that generalizes the factorial function. But nothing stops mathematicians from using "$\Gamma$" with other meanings, and they often do that.
$\Gamma$ is the uppercase form of the Greek letter gamma
It is common to refer to a function that has been named $ \phi $ as "$ \phi (x)$" (of course some other variable may be used instead of $x$). This is used with functions of more than one variable, too.
For functions given by formulas, this notation has the value of telling you what letter will be used for the input variable. Barred arrow notation also has this property.
This usage is very widespread, but strict writers would prefer "Let $h$ be a continuous function" and "The sine function is bounded". I recommend the strict practice: "$ \sin x $" is strictly speaking not the name of a function, but an expression denoting its value at $x$.
Abstractmath does not always follow this strict practice.
A function may be referred to by using its defining expression or defining equation. This is common in calculus books.
"The derivative of $ {{x}^{3}} $ is always nonnegative."
Very often the defining equation is used: "The derivative of $ y={{x}^{3}} $ is always nonnegative." If you analyze this example carefully, you see that it is literally nonsense. The equation $ y={{x}^{3}} $ is a statement. How can a statement have a derivative? Many mathematicians Frown Fiercely at this usage, but it is ubiquitous in math classrooms.
Using barred arrow notation ("the function $x\mapsto {{x}^{2}}+2x+5 $") or the defining expression ("the function $ {{x}^{2}}+2x+5 $ ") to refer to a function are two examples of anonymous notation for functions. This means no letter or word has been chosen to be a name for the function, which is desirable if you expect to refer to it only once or twice.
Another anonymous notation used in theoretical computing science is lambda notation, where you would refer to the same function as $\lambda x.{{x}^{2}}+2x+5 $. This usage is unfamiliar to most mathematicians outside computing science.
Anonymous notation is used for many kinds of math objects other than functions, for example setbuilder notation, and the usual notation for matrices.
In most math texts and on this website, the value of a function $ \phi $ at an input $x$ is written $ \phi (x) $. For example, if $ \phi $ is the squaring function, $ \phi (3)=9 $. (prefix notation, or "writing functions on the left"). But there are several other common ways to write the value of a function:
This section describes the major possibilities in some detail.
An expression is in prefix notation if the function symbols are written on the left of the input. This may be referred to as "writing functions on the left". This is the common way we write function values.
The traditional math symbolic language has certain conventions about prefix notation.
Don't confuse multiplication |
Pascal and many other computer languages require parentheses around all arguments to functions. Mathematica requires square brackets and in fact reserves square brackets for that use.
Infix notation is used for functions of two variables. You write the name of the function between the variables. Many functions denoted by symbols (as opposed to letters) are normally written this way, for example $x + y$ or $3/5$.
A special case of infix notation is juxtaposition or concatenation, which means writing nothing between two variables.
Multiplication has many notations:
Using postfix notation, you write the name of the function after its input. Most authors write functions of one variable in prefix notation, but some algebraists use postfix notation. Postfix notation may be called "writing functions on the right".
During the 1970's I wrote several papers using postfix notation. Many people complained. So I stopped doing so. On the other hand, the paper I wrote that got the most citations of my whole career was one of those papers. On the other other hand, at least three authors rewrote my proof$\ldots$
When the traditional infix notation is used for the basic operations of arithmetic, you have to use parentheses to distinguish between certain expressions. For example, $ a+bc $ and $\left( a+b \right)c $ give different values for most choices of numbers $a$, $b$, $c$.
When binary operations are written in prefix or postfix notation, you don't need parentheses. This is exhibited in the table below, in which I use $\ast $ for multiplication because the traditional juxtaposition notation doesn't work for prefix and postfix notation. (Think about it).
Infix |
Prefix |
Postfix |
$a + b \ast c$ |
$+ a \ast b\,\, c$ |
$a\,\, b\,\, c \ast +$ |
$(a + b)\ast c$ |
$\ast+ a\,\, b\,\, c$ |
$a\,\, b + c \ast$ |
$a\ast b + c$ |
$+\ast a\,\, b\,\, c$ |
$a\,\, b \ast c +$ |
$a \ast (b + c)$ |
$\ast\,\, a + b\,\, c$ |
$\,\,a\,\, b\,\, c + \ast$ |
Prefix notation without parentheses is called Polish notation and postfix notation without parentheses is called reverse Polish notation.
The programming language Lisp uses a form of Polish notation and the languages Forth and PostScript use reverse Polish notation exclusively. Most computer languages use infix notation, which computer people call algebraic notation.
Polish notation is named after the eminent Polish logician Jan Łukasiewicz, who invented the notation in the 1920's for use in logic. The terminology "reverse Polish notation" is a natural modification of this phrase and is not an ethnic slur.
A function is displayed in outfix notation (also called matchfix notation) if its symbol consists of characters (letters, digits and the like) or expressions put on both sides of the name of the input to the function (the argument). The pair of characters, one for each side, are called delimiters.
Functions like the integral whose inputs include functions are almost always called operators. Note that the variable $x$ does not appear in the list of arguments for Int. That is because it is a dummy variable.
In math English, those who use prefix notation (the usual notation) would say that the value of a function $f$ at an input $c$ ($f(c)$ in the symbolic language ) is "$f$ of $c$". We pronounce $\sin x$ as "sine of $x$".
Some very common functions have a more complicated naming system.
First we look at addition, subtraction, multiplication and division.
So if you add $3$ and $4$, you get $7$, which you say is the sum of $3$ and $4$. If you write the result of the addition, you write $3+4$ , which you pronounce "$3$ plus $4$". The table shows these details for the common arithmetic functions.
function |
verb (Note 1) |
symbol |
symbol name |
value |
addition |
add |
$+$ |
plus |
sum |
subtraction |
subtract |
$-$ |
minus |
difference |
multiplication |
multiply |
(Note 2) |
times |
product |
division |
divide |
(Note 3) |
(Note 3) |
quotient |
Note 1. These verbs use different prepositional phrases:
Note 2
Note 3
Note that for addition and subtraction, the name of the symbol is also the way you pronounce it in an expression: For example, "$+$" is pronounced "plus" and "$a+b$" is pronounced "a plus b".
In the remarks below, "shouldn't" means "if you do, people may look at you funny." Some students do say these things occasionally.
The expression "$a | b$" doesn't refer to a binary operation but I will mention it here anyway, since it causes enormous confusion in number theory and discrete math classes.
"$a/b$" is a number |
Differentiation is an operator that takes a function to its derivative.
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