example, the points (0,–3) and (0,3) on the circle of
radius three have the same first-coordinate value, and
the vertical line at x= 0 intercepts the graph of the cir-
cle at two locations. The circle is a relation, but it is
not a function.
A function f:X →Yis said to be “onto” (or “sur-
jective”) if every element of the codomain Yis the
image of some element of X. (That is, every element y
in Yis of the form y= f(x) for some xin X.) For exam-
ple, the squaring function thought of as a map from the
set {1,2,3,4} to the set {1,4,9,16} is onto. It is not onto,
however, when thought of as a map from the set of all
real numbers to the set of all real numbers: no negative
number results from the squaring operation.
208 function
History of Functions
Since the time of antiquity, scholars were interested in
identifying rules or relationships between quantities. For
example, the ancient Egyptians were aware that the cir-
cumference of a circle is related to its diameter via a fixed
ratio that we now call
PI
, and Chinese scholars, and later
the Pythagoreans, knew that the three sides of a right trian-
gle satisfy the simple relationship given by P
YTHAGORAS
’
S
THEOREM
. Although these results were not expressed in
terms of formulae and symbols (the evolution of algebraic
symbolism took many centuries), scholars were aware that
the value of one quantity could depend on the value of other
quantities under consideration. Although not explicit, the
notion of a “function” was in mind.
In the mid-1300s French mathematician N
ICOLE
O
RESME
discovered that a uniformly varying quantity (such as the
position of an object moving with uniform velocity, for
instance) could be represented pictorially as a “graph,” and
that the area under the graph represents the total change of
the quantity. Oresme was the first to describe a way of
graphing the relationship between an independent variable
and a dependent one and, moreover, demonstrate the use-
fulness of the task.
In 1694 German mathematician G
OTTFRIED
W
ILHELM
L
EIBNIZ
, codiscoverer of
CALCULUS
, coined the term function
(Latin: functio) to mean the
SLOPE
of the curve, a definition
that has very little in common with our current use of the
word. The great Swiss mathematician L
EONHARD
E
ULER
(1707–83) recognized the need to make the notion of a rela-
tionship between quantities explicit, and he defined the
term function to mean a variable quantity that is dependent
upon another quantity. Euler introduced the notation f(x) for
“a function of x,” and promoted the idea of a function as a
formula. He based all his work in calculus and
ANALYSIS
on
this idea, which paved the way for mathematicians to view
trigonometric quantities and logarithms as functions. This
notion of function subsequently unified many branches of
mathematics and physics.
In 1822 French physicist and mathematician J
EAN
B
AP
-
TISTE
J
OSEPH
F
OURIER
presented work on heat flow. He repre-
sented functions as sums of sine and cosine functions, but
commented that such representations may be valid only for
a certain range of values. This later led German mathemati-
cian P
ETER
G
USTAV
L
EJEUNE
D
IRICHLET
(1805–1859) to propose
a more precise definition:
A function is a correspondence that assigns a
unique value of a dependent variable to every
permitted value of an independent variable.
This, on an elementary level, is the definition generally
accepted today.
In the late 19th century, German mathematician G
EORG
C
ANTOR
(1845–1918) attempted to base all of mathematics on
the fundamental concept of a
SET
. Because the terms vari-
able and relationship are difficult to specify, Cantor pro-
posed an alternative definition of a function:
A function is a set of ordered pairs in which every
first element is different.
This idea is based on the fact that the graph of a function
is nothing more than a collection of points (x,y) with no
two y-values assigned to the same x-value. Cantor’s defi-
nition is very general and can be applied not only to num-
bers but to sets of other things as well. Mathematicians
consequently came to think of functions as “mappings”
that assign to elements of one set X, called the domain of
the function, elements of another set Y, called the
codomain. (Each element xof Xis assigned just one ele-
ment of Y.) One can thus depict a function as a diagram of
arrows in which an arrow is drawn from each member of
the domain to its corresponding member of the codomain.
The function is then the complete collection of all these
correspondences.
Advanced texts in mathematics today typically present
all three definitions of a function—as a formula, as a set of
ordered pairs, and as a mapping—and mathematicians will
typically work with all three approaches.
See also
GRAPH OF A FUNCTION
.