What is a Function?

We can think of a function as a machine that accepts certain objects, called inputs, and transforms them into other objects, called outputs. In order for our machine to be reliable and consistent, inserting two identical inputs should always result in the production of two identical outputs. If this is not the case, and two identical inputs are transformed by the machine into two different outputs, then there is some ambiguity as to the expected end product. This point is important because it distinguishes functions from relations. Functions defined on the real line accept real numbers (inputs) and assign each one exactly one real number (output). We call the set of acceptable inputs (i.e. the objects that the machine can accept) the domain of the function, and the set of corresponding outputs (i.e. the objects the machine outputs) the range of the function.

A Simple Example

Here is a simple example to help us better understand the concept of a function. Consider the following set of names:

{Lynn Acres, Dean Ford, Joel Rice, John Rowe, Anne Smythe, Vicky Smarte}.

Now consider the following set of initials:

{LA, DF, JR, AS, VS}.

Do you think the relation of assigning names to initials is a function? Since we are assigning names to initials, initials are the inputs and names are the outputs of our relation. Thus, the input LA results in the output Lynn Acres. What output corresponds to the input JR? It is clear after inspecting the set of names that the input JR outputs two different names, Joel Rice and John Rowe. Therefore, the relation of assigning names to initials is not a function because the input JR has 2 different outputs.

What happens if we reverse the relation and assign initials to names?

In this case, the names are the inputs to which initials are assigned . Does each name have one and only one corresponding initials? Yes, of course, by definition initials are assigned to one’s name uniquely. Thus, this relation is a function. Now that we have explored this simple example, we will extend these ideas using variable notation.

A mathematical example

In mathematics, it is customary to write a function using functional notation as follows,

y = f(x),

indicating that y is a function of x. In this case we think of the x-values as the inputs and the y-values as the outputs. We call the collection of all meaningful inputs (x) the domain of f, and the corresponding outputs (y) the range of f. In addition, we call y the dependent variable and x the independent variable since the value of y depends on x.

A note of caution

Remember, to determine if a relation is a function, we must be sure that each valid input corresponds to one and only one output. It is perfectly acceptable for two different inputs to correspond to only one output.

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In the next section we will demonstrate different ways to represent functions.

Function Representations