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The value of a function

Notation

# Notation

In most math texts and on this website, the value of a function  at an input x is written .  For example, if  is the squaring function, .  There are other ways to write the value, for example  or . When the function has two or more inputs, there are still other ways to write the value.  For example, we don’t usually write , we write

This section covers the major possibilities in use.

## Prefix notation

An expression is in prefix notation if the function symbols are written on the left of the input.

#### Examples

¨  This is the common way we write function values.  We write f(x) for an arbitrary function of one variable, and we write sin x, log x or log(x), and so on.  This includes functions of more than one variable if the function is written with a letter.  For example, let .  Then .

¨  As you know, we normally write the symbols for the arithmetic functions between the inputs, as in x + y.  See infix notation for more detail on this.

¨  The expression x + y written in prefix notation would be +(x,y).

¨  In fact the parentheses and comma are not necessary in the expression +(x,y).     + is a binary operation, so you can write + x y (see Polish notation).  This notation with spaces but no parentheses is discussed in more detail for postfix notation below.

¨  The programming language Lisp writes all its functions in prefix notation, but with yet another arrangement with parentheses and commas.  3 + 5 would be written (+ 3 5).

¨  Many functions whose names are special symbols have specific ways of writing expressions involving them.   Examples: x! (always postfix),  (outfix), x + y (usually infix),   (over the x),  (to the left and over the x).

### Parentheses around the input

¨  Most functions are written with the function named followed by the input name inside parentheses.  This includes most functions of two or more variables.

¨  It is customary to omit parentheses around the argument for trig functions such as “sin” and often for log functions.

¨  Many mathematical writers omit the parentheses in other situations too, writing “Fx” instead of “F(x)”.  Don’t  confuse evaluation written like this with multiplication. See Polish notation.

## Infix notation

Infix notation is used only for functions of two variables.  You write the name of the function between the variables.  The familiar operations +, , and / (division) are normally written this way, for example x + y or 3/5.

### Multiplication

Multiplication has many notations:

¨   as in  or  x  y.  But this symbol means vector product when put between 3-dimensional vectors.  More here.

¨  Juxtaposition as in xy, but only for variables, not for digits.

¨  Centered dot as in  or .

¨  Asterisk as in  or  (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 (except the blank space) between the arguments, but you have to mark it with tildes.

¨  Infix notation is also used for binary relations.

## 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 variables x and y as xy.   But “23” does not mean 2 times 3.

¨  It is standard to write the value of the sine function at x as sin x.  This does not mean the product of “sin” and x, which in fact is a meaningless idea since sin is not a number.

¨  If f and g are composable functions, the composite is commonly written gf.

## 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. The symbol "!" denoting the factorial function is normally written in postfix notation.

#### Example

The expression x+y in postfix notation is (x, y)+  or x y +.

You can understand the notation with spaces instead of parentheses and a comma this way:  In postfix notation the inputs come before (to the left) of the function.  So the x and y are the inputs and the function you apply to them is +.  You have to use spaces between the arguments, otherwise you get into trouble when the arguments are denoted by more than one letter or number.  For example, postfix notation for adding 65 and 34 would be 65 34 +.  If you wrote  6534+ you wouldn’t know where one argument ended and the other began.  When you used the traditional parentheses and comma notation, as in (65, 34)+, there is not a problem.

## 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,  and  give different values for most choices of numbers a, b, c.  This use of parentheses is distinct from the use to enclose the argument of a function.

When binary operations are written in prefix or postfix notation, you don’t need parentheses.  This is shown in the table.  In the table I use  for multiplication because the traditional juxtaposition notation doesn’t work for prefix and postfix notation.  (Think about it).

 infix prefix postfix

Prefix notation without parentheses is called Polish notation and postfix notation without parentheses is called reverse Polish notation.  The programming languages Forth and PostScript use reverse Polish notation exclusively.

Polish notation is named after the eminent Polish logician Jan ukasiewicz, 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 or expressions put on both sides of the argument.

#### Examples

¨  The notation (a, b) may denoted any one of several functions, discussed here.

¨  The absolute value of a number r is denoted by .

¨  The greatest integer in x is sometimes denoted by .  For example, .

¨  The notation (m, n) can mean the value of several different functions.

¨  Inner products on vector spaces are denoted by .  An alternative notation is

## Other notation styles

¨  The way we write the definite integral can be seen as a fancy way of writing a function of three variables.  The integral  is a function that takes two numbers a and b and a function f  and gives you a number.  For example, .  (Functions like this whose inputs include functions are almost always called operators.)

¨  Any list  can be thought of as a function on its index set.   For example, the ordered pair (3, 5) can be regarded as the function whose value at 1 is 3 and whose value at 2 is 5.  Some authors use this point of view without comment.  They say things like:

The sequence  with  is decreasing in n.

¨  may also be written as subscripts.  For example you can refer to the one-parameter family of functions  defined by .    For each number a,  is a function of one variable x.

### Note on terminology

The “fix” terminology comes from computing science, and some mathematicians also use it.  Others, instead of saying, “I use postfix notation”, will say, “I write my functions on the right,” and so on.

# Name and value

At the beginning of this page I discussed how to write the value of a function in the symbolic language.  In math English,  those who use prefix 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 would pronounce sin x as “sine of x”.

Some very common functions have a more complicated naming system.

¨  The value of the function at an input has another name.  Example: “The sum of c and d” (not “the addition of c and d.”

In some cases, there is a standard symbol for the function and this symbol has a third name.  Example: the symbol + for addition.  The name of this symbol is “plus”.

So if you add 3 and 5, you get 8, which you say is the sum of 3 and 5.  If you write it in symbols, you write , which you pronounce “3 plus 5”.

The table shows the common functions for which this happens.  Note that both differentiation and integration involve several different symbolic notations, not shown here.

 function verb symbol symbol name value addition add plus sum subtraction subtract minus difference multiplication multiply times division divide divided by quotient squaring square (note 1) squared square composition compose composite differentiation differentiate derivative integration integrate integral

Note 1.  The symbol for squaring is a postscripted superscript 2, as in

### Remarks

In the remarks below, “shouldn’t” means “if you do, people may look at you funny.”  See how languages change.  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” or “the differentiation of f”.

¨  You shouldn’t use the symbol name for the result.  For example, you shouldn’t say "the plus of 3 and 5".

¨  The names "plus", "minus" and "times" may be used with "sign" to name the symbol directly (the plus sign, the minus sign).

¨  The symbol "  " is traditionally called the "division sign", but I notice younger people call it the "divided by" sign.

### Composition

¨  The value of the composite  at an input x may be written  or gf(x)  or (most commonly) g(f(x)).

¨  When you pronounce  you can say " g composed with f" or "the composite of g and f”.

¨  Many writers blur the distinction between “composition” and “composite” and refer to  as the "composition" of g and f.  I personally hate this usage, but see how languages change.

The chapter on composition has more about all this.