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Last edited 4/14/2009 10:37:00 AM
The value of a function

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.
An expression is in prefix notation if the function symbols are
written on the left
of the input.
¨ 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).
The traditional math symbolic
language has certain conventions about prefix notation.
¨ 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 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 has many
notations:
¨ as in or x y. But
this symbol means vector product when put between 3dimensional 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).
¨ 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.
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.
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.
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.
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.
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.
¨ 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
¨ 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.
¨ Parameters may also be written as subscripts. For example you can refer to the oneparameter family of functions defined by . For each number a, is a function of one variable x.
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.
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 function has a name. Example: “addition”, corresponding verb “add”.
¨ 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 



differentiation 
differentiate 


derivative 
integration 
integrate 


integral 
Note 1. The symbol for squaring is a postscripted superscript 2, as in
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
¨ 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.
¨ 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.
See also Notation for Functions.