## Define a Function

### Learning Outcomes

• Define a function using tables
• Define a function from a set of ordered pairs
• Define the domain and range of a function given as a table or a set of ordered pairs

There are many kinds of relations. A relation is simply a correspondence between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its U.S. senators. Each state can be matched with two individuals who have each been elected to serve as a senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.

The first value of a relation is an input value and the second value is the output value. A function is a specific type of relation in which each input value corresponds to one and only one output value. An input is called the independent value, and an output is  called the dependent value, as it depends on the value of the input.

Notice in the first table below, where the input is “name” and the output is “age,” each input matches with exactly one output. This is an example of a function.

Family Member’s Name (Input) Family Member’s Age (Output)
Nellie $13$
Marcos $11$
Esther $46$
Samuel $47$
Nina $47$
Paul $47$
Katrina $21$
Andrew $16$
Maria $13$
Ana $81$

Compare this with the next table where the input is “age” and the output is “name.” Some of the inputs result in more than one output. This is an example of a correspondence that is not a function.

Family Member’s Age (Input) Family Member’s Name (Output)
$11$ Marcos
$13$ Nellie, Maria
$16$ Andrew
$21$ Katrina
$46$ Esther
$47$ Samuel, Nina, Paul
$81$ Ana

Now let us look at some other examples to determine whether the given relations are functions, and under what circumstances. Remember that a relation is a function if there is only one output for each input.

### Example

Fill in the table.

Input Output Function? Why or why not?
Name of senator Name of state
Name of state Name of senator
Time elapsed Height of a tossed ball
Height of a tossed ball Time elapsed
Number of cars Number of tires
Number of tires Number of cars

Relations can be written as ordered pairs of numbers $(x, y)$ or as numbers in a table of values the columns of which each contain inputs or outputs. By examining the inputs (x-coordinates) and outputs (y-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input corresponds to only one output. That is, each value corresponds to exactly one y value.

There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the domain of the function. The set of output values is called the range of the function.

Given a set of ordered pairs, find the domain by listing all of the input values, which are the x-coordinates. To find the range, list all of the output values, which are the y-coordinates.

Consider the following set of ordered pairs:

$\{(−2,0),(0,6),(2,12),(4,18)\}$

You have the following:

$\begin{array}{l}\text{Domain:}\{−2,0,2,4\}\\\text{Range:}\{0,6,12,18\}\end{array}$

Now try it yourself.

### Example

List the domain and range for the following table of values where x is the input and y is the output.

x y
$−3$ $4$
$−2$ $4$
$−1$ $4$
$2$ $4$
$3$ $4$

Watch the following video for another example of how to identify whether a table of values represents a function and how to determine the domain and range.

### Example

Define the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.

$\{(−3,−6),(−2,−1),(1,0),(1,5),(2,0)\}$

The following video shows how to determine whether a relation given in non-tabular form is a function and how to find the domain and range.

### Example

Find the domain and range of the relation and determine whether it is a function.

$\{(−3, 4),(−2, 4),( −1, 4),(2, 4),(3, 4)\}$

## Summary: Determining Whether a Relation is a Function

1. Identify the input values – this is your domain.
2. Identify the output values – this is your range.
3. If each value in the domain leads to only one value in the range, classify the relationship as a function. If any value in the domain leads to two or more values in the range, do not classify the relationship as a function.