We will frequently use the formal concept of a set, which is just a collection of
objects, called elements. Examples of sets include the real numbers
**R**
, the
integers, the set of names of the days in a week, and the set of letters in the
alphabet. One kind of set that we will encounter fairly often is called an
interval. The open interval
(*a*, *b*)
consists of the real numbers
*x*
such that
*a* < *x* < *b*
, while the closed interval
[*a*, *b*]
consists of the real numbers
*x*
such that
*a*≤*x*≤*b*
. If
*x*
is an element of the set
*S*
, we write
*x*âàà*S*
. Thus
*Π*âàà*realnumbers*
,
1âàà(0, 2)
, and Tuesday
âàà
\. A function
*f*
from a set
*S*
to a set
*T*
is a rule that takes an
element of the set
*S*
and gives back an element of the set
*T*
. We denote this by
*f* : *S*→*T*
. The set
*S*
is called the domain of the function
*f*
and the
set
*T*
is called its range.

Suppose we have a function
*f* : *S*→*T*
, with
*x*âàà*S*
. If
*f*
takes an element
*x*âàà*S*
to
*y*âàà*T*
, we write
*f* : *x*
*y*
or
*f* (*x*) = *y*
, and say that "
*f*
maps
*x*
to
*y*
." We often call this element
*y*
the image of
*x*
under
*f*
, and
denote it by
*f* (*x*)
. This is illustrated in the figure below.

Figure %: Plot of a Function
*f* : *S*→*T*

If
*f* : *S*→*T*
and
*g* : *T*→*U*
, then we can define a new function
*g*
o
*f* : *S*→*U*
by
(*g*
o
*f* )(*x*) = *g*(*f* (*x*))
for each element
*x*âàà*S*
. The
function
*g*
o
*f*
is called the composition of the functions
*g*
and
*f*

The graph of a function is the set of all points of the form
(*x*, *f* (*x*))
. One can draw
this by plotting points on a pair of coordinate axes, with the horizontal axis
corresponding to
*x*
, and the vertical corresponding to
*f* (*x*)
.

A function
*f* : *S*→*T*
is called invertible if there exists a function
*g* : *T*→*S*
such that
(*g*
o
*f* )(*x*) = *x*
for each element
*x*âàà*S*
. If
*f*
is
invertible, then this function
*g*
is called the inverse of
*f*
. One way to tell if a
function is invertible is to look at its graph. A function is invertible if and only
if no horizontal line intersects the graph in more than one point. Take a moment to
convince yourself that this is true.

(1) The most familiar functions map the set of real numbers to itself. That is,
*f* : **R**→**R**
. An example is the function
*f*
such that for each
real number
*x*
,
*f* (*x*) = 2*x*
, i.e. the image of each element
*x*
is the element
2*x*
.
We may graph this function as follows: