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# 3.3 Functions

3.37   Definition (Function.) Let be sets. A function with domain and codomain is an ordered triple where is a rule which assigns to each element of a unique element of . The element of which assigns to an element of is denoted by . We call the -image of or the image of under . The notation is an abbreviation for  is a function with domain and codomain ''. We read  '' as  is a function from to .''

3.38   Examples. Let be defined by the rule

Then is not defined, because

Let be defined by the rule: for all

Thus , , ,

3.39   Definition (Maximum and minimum functions.) We define functions max and min from to R by the rule
 (3.40)

 (3.41)

Thus we have

and

for all . Also

3.42   Definition (Absolute value function.) Let be defined by the rule

We call the absolute value function and we usually designate by .

3.43   Definition (Sequence) Let be a set. A sequence in is a function . I will refer to a sequence in as a real sequence.

The sequence is sometimes denoted by . Thus is the sequence such that for all . Sometimes the sequence is denoted by
 (3.44)

for example is the same as . The notation in formula  (3.44) is always ambiguous. I will use it for sequences like

in which it is somewhat complicated to give an analytic description for .

If is a sequence, and , then we often denote by .

3.45   Examples. Let denote the set of all polygons in the plane. For each number in let

For each let

and

denote the polygons inscribed in and containing described in section2.1. Then

and are sequences in .

is a real sequence. (Cf. (2.3) and (2.12).)

is a sequence of intervals.

3.46   Definition (Equality for functions.) Let and be two functions. Then, since a function is an ordered triple, we have

The rules and are equal if and only if = for all . If and then it is customary to write to mean . This is an abuse of notation, but it is a standard practice.

3.47   Examples. If is defined by the rule

and is defined by the rule

then since and have different codomains.

If and are defined by the rules

then .

In certain applications it is important to know the precise codomain of a function, but in many applications the precise codomain is not important, and in such cases I will often omit all mention of the codomain. For example, I might say For each positive number , let .'' and proceed as though I had defined a function. Here you could reasonably take the codomain to be the set of real intervals, or the set of closed intervals, or the set of all subsets of R.

3.48   Definition (Image of ) Let be sets, and let . The set

is called the image of , and is denoted by More generally, if is any subset of then we define

We call the -image of . Clearly, for every subset of we have .

3.49   Examples. If is defined by the rule

then

.

3.50   Definition (Graph of ) Let be sets, and let . The graph of is defined to be

If the domain and codomain of are subsets of R, then the graph of can be identified with a subset of the plane.

3.51   Examples. Let be defined by the rule

The graph of is sketched below. The arrowheads on the graph are intended to indicate that the complete graph has not been drawn.

Let Let be the function from to R defined by the rule

The graph of is sketched above. The solid dot at indicates that is in the graph. The hollow dot at indicates that is not in the graph.

Let be defined by the rule

Thus and The graph of is sketched above.

The term function (functio) was introduced into mathematics by Leibniz [33, page 272 footnote]. During the seventeenth century the ideas of function and curve were usually thought of as being the same, and a curve was often thought of as the path of a moving point. By the eighteenth century the idea of function was associated with analytic expression''. Leonard Euler (1707-1783) gave the following definition:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.

Hence every analytic expression, in which all component quantities except the variable are constants, will be a function of that ; Thus ; ; ; ; etc. are functions of [18, page 3].

The use of the notation '' to represent the value of at was introduced by Euler in 1734 [29, page 340].

3.52   Exercise. Sketch the graphs of the following functions:

a)
.
b)
.
c)
.
d)
.

Next: 3.4 Summation Notation Up: 3. Propositions and Functions Previous: 3.2 Sets Defined by   Index
Ray Mayer 2007-09-07