A function (or map) is a rule or correspondence that
associates each element of a set X called the domain with a
unique element of another set Y called the codomain. We typically
give the rule a name such as a letter like f or g (or any letter of your
choice) or a name agreed upon by convention like sine or log or square root.
The term ''unique'' is critical in the definition as this says that one 
cannot be associated with 2 or more elements of Y. Thus we can
call the y value corresponding to a particular x under the rule f by
the name f(x) (read f of x) since there is only one such y. However,
associating 2 elements of X with the same 
and some elements of Y
not corresponding to any element of X are allowed as they are not ruled
out in the definition. The fact that the correspondence goes from the domain
to the codomain is indicated by the notation 
.
For a subset 
we use f(A) to mean the set 
called the image of A. (Some
authors use 
.) The set f(X) is called the range of f. While it is always the case that 
, in general
they need not be equal. It is important to distinguish while 
.
For a subset 
we use 
to mean the set 
called the inverse image of B. (Some authors use
.) Keep in mind that while 
.
If and 
we can form a new function
called a composite map denoted by 
. The
symbol 
is read as ''g circle f.'' The composition is
accomplished by defining 
for each .
is called an injection or one-to-one (also written 1-1) if different x values get mapped to
different y values. So if 
then 
. this is
equivalent to the contrapositive, if 
then 
. We call
f a surjection or onto if every corresponds to
some , that is, f(X)=Y. A bijection is used for
functions that are both 1-1 and onto.
The set of ordered pairs 
is the
graph of f.