We define the concept of a function.

**the position of an object with respect to time.**

Our observations seem to indicate that every instant in time is associated to a unique positioning of the objects in the universe. You may have heard the saying,

**you cannot be two places at the same time,**

and it is this fact that motivates our definition for functions.

Something as simple as a dictionary could be thought of as a relation, as it connects
*words* to *definitions*. However, a dictionary is not a function, as there are words with
multiple definitions. On the other hand, if each word only had a single definition,
then a dictionary would be a function.

- Since words may have more than one definition, “relating words to their definition in a dictionary” is not a function.
- Since for any given time, objects cannot be “two places at once,” this is a function.
- Since every person only has one birth date, “relating people to their birth date” is a function.
- Since mothers can have more (or less) than one child, “relating mothers to their children” is not a function.

What we are hoping to convince you is that the following are true:

- (a)
- The definition of a function is well-grounded in a real context.
- (b)
- The definition of a function is flexible enough that it can be used to model a wide range of phenomena.

Whenever we talk about functions, we should explicitly state what type of things the inputs are and what type of things the outputs are. In calculus, functions often define a relation from (a subset of) the real numbers (denoted by ) to (a subset of) the real numbers.

**domain**, and we call the set of the outputs of a function the

**range**.

*described*by the formula or by the graph shown in the plot below:

- A formula
**describes**the relation using symbols. - A graph
**describes**the relation using pictures.

The **function is the relation itself**, and is independent of how it is described.

Our next example may be a function that is new to you. It is the *greatest integer
function*.

**greatest integer function**. This function maps any real number to the greatest integer less than or equal to . People sometimes write this as , where those funny symbols mean exactly the words above describing the function. For your viewing pleasure, here is a graph of the greatest integer function:

Notice that both the functions described above pass the so-called *vertical line
test*.

**vertical line test**.

Sometimes the domain and range are the *entire* set of real numbers, denoted by . In
our next examples we show that this is not always the case.

To really tease out the difference between a function and its description, let’s consider an example of a function with two different descriptions.

Finally, we will consider a function whose domain is all real numbers except for a single point.

Since these two functions do not have the same graph, and they do not have the same domain, they must not be the same function.

However, if we look at the two functions everywhere except at , we can say that . In other words, From this example we see that it is critical to consider the domain and range of a function.