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# Defining a Function

### Learning Outcomes

• Define a function using tables
• Determine if a set of ordered pairs creates a function
• 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 has one and only one output value. An input is the independent value, and the output is 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 relations are functions or not 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

Input Output Function? Why or why not?
Name of senator Name of state Yes For each input, there will only be one output because a senator only represents one state.
Name of state Name of senator No For each state that is an input, 2 names of senators would result because each state has two senators.
Time elapsed Height of a tossed ball Yes At a specific time, the ball has one specific height.
Height of a tossed ball Time elapsed No Remember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.
Number of cars Number of tires Yes For any input of a specific number of cars, there is one specific output representing the number of tires. (note: this would be assuming that all cars have four tires!)
Number of tires Number of cars Yes For any input of a specific number of tires, there is one specific output representing the number of cars.

Relations can be written as ordered pairs of numbers or as numbers in a table of values. 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 has only one output. 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. If you have a set of ordered pairs, you can 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: [latex-display]\{(−2,0),(0,6),(2,12),(4,18)\}[/latex-display] You have the following: [latex-display]\begin{array}{l}\text{Domain:}\{−2,0,2,4\}\\\text{Range:}\{0,6,12,18\}\end{array}[/latex-display] 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
$−3$ $4$
$−2$ $4$
$−1$ $4$
$2$ $4$
$3$ $4$

Answer: The domain describes all the inputs, and we can use set notation with brackets { } to make the list. [latex-display]\text{Domain}:\{-3,-2,-1,2,3\}[/latex-display] The range describes all the outputs. [latex-display]\text{Range}:\{4\}[/latex-display] We only listed $4$ once because it is not necessary to list it every time it appears in the range.

In the following video we provide another example of identifying whether a table of values represents a function as well as determining the domain and range of each. [embed]https://youtu.be/y2TqnP_6M1s[/embed]

### Example

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

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

Answer: We list all of the input values as the domain.  The input values are represented first in the ordered pair as a matter of convention. Domain: {$-3,-2,1,2$} Note how we did not enter repeated values more than once; it is not necessary. The range is the list of outputs for the relation; they are entered second in the ordered pair. Range: {$-6, -1, 0, 5$} Organizing the ordered pairs in a table can help you tell whether this relation is a function.  By definition, the inputs in a function have only one output.

x y
$−3$ $−6$
$−2$ $−1$
$1$ $0$
$1$ $5$
$2$ $0$
The relation is not a function because the input $1$ has two outputs: $0$ and $5$.

In the following video, we show how to determine whether a relation is a function and how to find the domain and range. [embed]https://youtu.be/kzgLfwgxE8g[/embed]

### Example

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

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

Answer: Domain: {$-3, -2, -1, 2, 3$} Range: {$4$} To help you determine whether this is a function, you could reorganize the information by creating a table.

x y
$−3$ $4$
$−2$ $4$
$−1$ $4$
$2$ $4$
$3$ $4$
Each input has only one output, and the fact that it is the same output (4) does not matter. This relation is a function.

## 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.

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