Functions: Notation and Terminology

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# FUNCTIONS: NOTATION AND TERMINOLOGY

## Terminology for functions

### Other words meaning "function"

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.)

#### Map, mapping

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.

#### Transformation

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.

#### Operator

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.

#### Functional

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.

#### Operation

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.

### Arrow Notation

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.

#### Pronunciation of arrow notation

The expression "$f :A\to B$" can be used as a name or as an independent sentence.

• Standing alone, the expression may be read aloud this way: "$f$ is a function from A to B" or "$f$ goes from $A$ to $B$".
• The expression may occur embedded in a sentence, as in "Let $f :A\to B$ be a differ­entiable function." This may be read "Let $f$ from $A$ to $B$ be a differ­entiable function."
##### Example:

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.

## Ways of defining a function

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.

### Table

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.

### Formula

A function may be defined by giving an algebraic expression (its formula) that determines its value. An example is the function $g$ defined by $g(x)={{x}^{2}}+2x-4$.

• The expression "$g(x)={{x}^{2}}+2x-4$" may be called the defining equation of the function, and "${{x}^{2}}+2x-4$" its defining expression.
• The defining expression may be used as its name as in "${{x}^{2}}+2x-4$ has one local minimum".
• In a calculus book you can assume the domain is $\mathbb{R}$. In a complex analysis text its domain could be $\mathbb{C}$, but in that text it would probably be written "$g(z)={{z}^{2}}+2z-4$", using the variable $z$. That is by convention.
• Two functions given by differ­ent formulas may be the same function. If you define $f:\mathbb{R}\to\mathbb{R}$ by $f(x)=x^2+2x-4$ and $g:\mathbb{R}\to\mathbb{R}$ by $g(x)=(x+1)^2-5$ then $f$ and $g$ are the same function.

### Algorithm

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).

• The program may fail or run forever on some inputs. This is true of Newton's method.
• Are two programs in differ­ent languages that "do the same thing" the "same algorithm"?
• Are two programs that give the same output for each input the "same algorithm"?
• "Give the same output" and "do the same thing" are not the same thing!

### Geometric definition

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).

### Barred arrow notation

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.

##### Example:

"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!

#### Use with parameters

Using the barred arrow clears up ambiguity when the defining expression has parameters in it.

##### Example:

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 $(x\mapsto {{x}^{2}}+ax+b):\mathbb{R}\to \mathbb{R}$. "

#### With finite functions

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: $1\mapsto 3,\,\,\,2\mapsto 3,\,\,\,3\mapsto 2,\,\,\,6\mapsto 1$.

## The name of a function

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".

### Naming a function by a letter

A function may be named by a letter of some alphabet, for temporary use in that particular section of text.

##### Examples:
• A paragraph may begin, "Let $f:\mathbb{R}\to\mathbb{R}$ be the squaring function", then the name of the function in that paragraph (or perhaps in that section) is $f$. The value of the function at $3$ is denoted by "$f(3)$", which in this example is $9$.
• "Let $\phi :\mathbb{R}\to \mathbb{R}$ be a continuous function with no derivative at 0." In this case, the letter $\phi$ (phi) does not refer to a specific function such as the squaring function, but an unspecified function that is subject to the constraint that it has no derivative at $0$.
• "Let $\psi (x)={{x}^{2}}-1$ ". $\psi$ is the letter psi. This names the function and simultaneously gives a formula for calculating its value.

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.

### Global names

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.

##### Example:

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

### Naming a function by its value at  $x$

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.

##### Examples:
• "Let $h(x)$ be a continuous function."
• "$\sin x$ is periodic."

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.

### Formula or equation as name of a function

A function may be referred to by using its defining expression or defining equation. This is common in calculus books.

##### Example:

"The derivative of ${{x}^{3}}$ is always nonnegative."

##### Example:

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.

### Anonymous notation

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.

## Notation for the value of a function

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:

• Factorial function: $x!$ (postfix notation, or "writing functions on the right").{/li>
• Absolute value: $\left| x \right|$ (outfix notation).
• Addition: $x + y$ (infix notation).
• Time derivative: $\dot{x}$ (over the $x$).
• Square root: $\sqrt{x}$ (to the left and over the $x$).

This section describes the major possibilities in some detail.

### Prefix notation

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.

#### Parentheses around the input

• Most functions are written with the function named followed by the input name inside parentheses.
• It is customary to omit parentheses around the argument for trig functions such as "sin" and often for log functions, so we may write "$\sin \pi$" or "$\log 2$".
• This includes functions of two or more variables when the function name is written with a letter, as in "$f(3,5)$". For example, let $f(x,y)={{x}^{2}}+2{{y}^{2}}$. Then $f(3,5)=59$. Note that $f(3,5)\neq f(5,3)$.
• Many mathematical writers omit the parentheses in other situations too, writing "fx" instead of "f(x)". In their notation, "$fx$" does not mean "$f$ times $x$" and "$\sin x$" does not mean "$\sin$ times $x$". (This is Polish notation.)
 Don't confuse multiplication with prefix notation without parentheses.

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

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 $3/5$.

• The expression "$x + y$" written in prefix notation would be "$+(x,y)$".
• $(x+y)+z$ would be $+(+(x,y),z)$ and $x+(y+z)$ would be $+(x,+(y,z))$. Since addition is associative, $(x+y)+z=x+(y+z)$, which in infix notation is normally written $x+y+z$.
• In many programming languages that use prefix notation, you may write $+(x,y,z)$ and similarly for other associative operations
• Subtraction is not associative, and as a result an expression such as $x-y-z$ is ambiguous; it must be written either $(x-y)-z$ or $x-(y-z)$.

#### Juxtaposition

A special case of infix notation is juxtaposition or concatenation, which means writing nothing between two variables.

• In standard algebraic notation, we write the product of numbers $x$ and $y$ as "$xy$". This is used only for variables represented by single letters, not digits: "23" does not mean 2 times 3.
• If $f$ and $g$ are composable functions, the composite is commonly written "$gf$".
• In an algebraic structure, a binary operation may be written by juxtaposition. This is the default notation for the multiplication in a ring or field, and it is the most common notation for the binary operation that defines a monoid or a group.

#### Multiplication

Multiplication has many notations:

• $\times$ as in $3\times 5$ or x $\times$ y. But this symbol means vector product when put between 3-dimensional vectors. More here
• Juxtaposition, as mentioned above.
• Centered dot as in "$3\cdot 5$" or "$x\cdot y$".
• Asterisk as in "$3\star 5$" or "$x\star y$" (mostly in programming languages). Variables in programming languages tend to have multiletter names, and juxtaposition doesn't work with them.
• Blank space, as in "3 5" or "$x\ y$" (in Mathematica).

#### Notes

• Mathematica allows you to write any function of two variables between the arguments, but if its name is a string made up of letters, you have to mark it with tildes
• Infix notation is also used for binary relations.

### Postfix notation

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".

##### Examples
• The expression $f(x)$ (which is in prefix notation) would be written $(x)f$ or $xf$ in postfix notation. The version without parentheses is much more common.
• The symbol "!" denoting the factorial function is normally written in postfix notation.
• The expression $x+y$ in postfix notation is $(x, y)+$

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$\ldots$

#### Polish notation

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.

### Outfix notation

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.

##### Examples
• The absolute value of a number $r$ is denoted by $\left| r \right|$. The two delimiters are identical vertical lines.
• Inner products on vector spaces may be denoted by $\left\langle v,\,\,w \right\rangle$.
• The notation $(a, b)$ may denote any one of several functions, discussed in the abmath article on delimiters.. The delimiters are "(" (left) and ")" (right).
• The "greatest integer in a real number $x$", or the floor of $x$, is sometimes denoted by $\left\lfloor x \right\rfloor$. For example,$\left\lfloor \pi \right\rfloor =3$ and $\left\lfloor -1.5 \right\rfloor =-2$ (not $-1$!).

### Other notation styles

• The bra-ket notation for the inner product of two vectors $v$ and $w$ is "$\left\langle v | w \right\rangle$". This combines infix and outfix notation.
• The definite inte­gral $\int_{a}^{b}f(x)\,dx$ is a function that takes two num­bers $a$ and $b$ and a function $f$ and gives you a number. So the way we write the definite integral can be seen as a fancy way of writing a function of three variables. If we call this function "Int" and write it in prefix notation, we could say $\text{Int}(a,b,f)=\int_{a}^{b}{f(x)\,dx}$ For example, $\text{Int}(0,\pi,\text{sin}) =\int_{0}^{\pi }{\sin \,x\,dx}=2$
• 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.

• Any list $\left( {{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,... \right)$ can be thought of as a function on its index set. For example, the ordered triple $\left( {{a}_{1}},\,{{a}_{2}},\,{{a}_{3}}\right)=(3, 5,0)$ has index set $\{1,\,2,\,3\}$, and thought of as a function it is $f:\{1,2,3\}\to\{0,3,5\}$ defined by $f(1):=3$, $f(2):=5$, $f(3):=0$. In barred arrow notation the function is $1\mapsto 3, 2\mapsto 5, 3\mapsto 0$

## Name and value

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.

### Common arithmetic operations

First we look at addition, subtraction, multiplication and division.

• The procedure for carrying out the operation has a name, for example addition.
• The verb that describes carrying out the operation may be a differ­ent word. For example, we "add $3$ and $4$" to get $7$.
• The value of the function at an input has another name. For example we say, "The sum of 3 and $4$ is $7$" (not "the addition of $3$ and $4$.")
• The operation is denoted by one or more symbols. For addition, the symbol is "$+$".
• The symbol has a name. The name of the symbol "$+$" is plus.

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

#### Notes

Note 1. These verbs use different prepositional phrases:

• "Add $2$ and $3$" or "Add $2$ to $3$" means "$2+3$".
• "Subtract $2$ from $3$" means $3-2$ (note the reversal).
• "Multiply $2$ by $3$" means $2\times 3$.
• "Divide $2$ by $3$" means $2/3$ or $3\div 2$.
• But "Divide $2$ into $3$ means $3/2$ or $2\div 3$ (!).

Note 2

• Multiplication may be denoted by the asterisk as in $a\ast b$, by concatenation as in $ab$ or by the centered dot as in $a\cdot b$.
• The expressions "$a\cdot b$" and "$a\times b$" are both pronounced "a times b" when the symbol denotes numerical multiplication or multiplication of matrices.
• The expression "$a\cdot b$" is pronounced "a dot b" when $a$ and $b$ are vectors, and "$a\times b$" is pronounced "a cross b" when $a$ and $b$ are three-dimensional vectors.
• Both symbols are used with other meanings, in some cases with other pronunciations, as well.

Note 3

• Either symbol "$/$" or "$\div$" may be called "the division symbol".
• Either phrase "$a/b$" or "$a\div b$" is pronounced "a divided by b", and "$a/b$" is also pronunced "a over b".
• The symbol "$\div$" as far as I know does not have a common name like "plus" for "$+$". The symbol "$/$" is often called slash, but usually "$a/b$" is pronounced "a over b".

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".

#### Remarks

In the remarks below, "shouldn't" means "if you do, people may look at you funny." Some students do say these things occasionally.

• You shouldn't use the operation name for the result. For example, you shouldn't say "the addition of 3 and 5" (say "sum") or "the differ­entiation of f" (say "derivative".
• You shouldn't use the symbol name for the result. For example, you shouldn't say "the times of 3 and 5" (say "product").

### Don't lean on a vertical line

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 sentence, meaning "a divides b with no remainder" and is used only for integers.
• In other words, "$a | b$" means $b/a$ is an integer.
##### Examples
• "$3|6$" is true because $6/3=2$, an integer.
• "$3|5$" is false because $5/3$ is not an integer.
 "$a/b$" is a number "$a|b$" is a statement which is either true or false.

#### Composition of functions

• The composite of functions $f:A\to B$ and $g:B\to C$ may be written $(g\circ f)$ or $gf$. The value at an input $x$ is most commonly written $g(f(x))$.
• When you pronounce $g\circ f$ you can say "$g$ composed with $f$" or "the composite of $g$ and $f$". It really means "do $f$ then $g$". I have heard people say "$f$ then $g$" or "$f$ followed by $g$".
• Watch out: $gf(x)$ can mean either $(g\circ f)(x))$ or $g(x)f(x)$ in a situ­ation where elements in the domain and codomain can be multiplied (for example if they are real numbers). For example, if $f(x)=x+1$ and $g(x)=x^2$, then $gf(x)$ can mean either $(x+1)^2$ (the composite) or $x^2(x+1)$ (the product).
• Also there is a horrendous problem with people who want to write $f\circ g$ instead of $g\circ f$. Wikipedia dis­cusses this.
• Many writers blur the distinction between "composition" and "composite" and refer to $g\circ f$ as the "composition" of $g$ and f. I personally hate this usage, but see how languages change.

### Differ­entiation

Differ­entiation is an operator that takes a function to its deriv­ative.

• The name of the operation is "differ­entiation".
• The verb is "differ­entiate", but you can also say "take the deriv­ative of".
• The result of differ­entiating $f$ is called the deriv­ative of $f$.
• As you probably know, there are bunches of ways of writing it. $Df$, $\frac{df}{dx}$, and $f'$ are some of them.
• If $x$ is a function of $t$ you can write the deriv­ative of $x$ with respect to $t$ as "$\dot{x}$".