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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:

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