We define the concept of a function.

Life is complex. Part of this complexity stems from the fact that there are many relationships between seemingly unrelated events. Armed with mathematics, we seek to understand the world. Perhaps the most relevant “real-world” relation is

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.

If our function is the “position with respect to time” of some object, then the input is
position time none of the above
and the output is
position time none of the above

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.

Which of the following are functions?
Mapping words to their definition in a dictionary. Given an object traveling through space, mapping time to the position of the object in space. Mapping people to their birth date. Mapping mothers to their children.

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

The definition of a function is well-grounded in a real context.
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.

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


Notice that both the functions described above pass the so-called 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.