# 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

**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 | [latex]13[/latex] |

Marcos | [latex]11[/latex] |

Esther | [latex]46[/latex] |

Samuel | [latex]47[/latex] |

Nina | [latex]47[/latex] |

Paul | [latex]47[/latex] |

Katrina | [latex]21[/latex] |

Andrew | [latex]16[/latex] |

Maria | [latex]13[/latex] |

Ana | [latex]81[/latex] |

*not*a function.

Family Member's Age (Input) | Family Member's Name (Output) |
---|---|

[latex]11[/latex] | Marcos |

[latex]13[/latex] | Nellie, Maria |

[latex]16[/latex] | Andrew |

[latex]21[/latex] | Katrina |

[latex]46[/latex] | Esther |

[latex]47[/latex] | Samuel, Nina, Paul |

[latex]81[/latex] | Ana |

*output for each input.*

**one**### 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 |

Answer:

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. |

*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 |
---|---|

[latex]−3[/latex] | [latex]4[/latex] |

[latex]−2[/latex] | [latex]4[/latex] |

[latex]−1[/latex] | [latex]4[/latex] |

[latex]2[/latex] | [latex]4[/latex] |

[latex]3[/latex] | [latex]4[/latex] |

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 [latex]4[/latex] once because it is not necessary to list it every time it appears in the range.

### Example

Define the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.[latex]\{(−3,−6),(−2,−1),(1,0),(1,5),(2,0)\}[/latex]

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: {[latex]-3,-2,1,2[/latex]} 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: {[latex]-6, -1, 0, 5[/latex]} 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 |
---|---|

[latex]−3[/latex] | [latex]−6[/latex] |

[latex]−2[/latex] | [latex]−1[/latex] |

[latex]1[/latex] | [latex]0[/latex] |

[latex]1[/latex] | [latex]5[/latex] |

[latex]2[/latex] | [latex]0[/latex] |

### Example

Find the domain and range of the relation and determine whether it is a function.[latex]\{(−3, 4),(−2, 4),( −1, 4),(2, 4),(3, 4)\}[/latex]

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

x |
y |
---|---|

[latex]−3[/latex] | [latex]4[/latex] |

[latex]−2[/latex] | [latex]4[/latex] |

[latex]−1[/latex] | [latex]4[/latex] |

[latex]2[/latex] | [latex]4[/latex] |

[latex]3[/latex] | [latex]4[/latex] |

**Summary: Determining Whether a Relation is a Function**

- Identify the input values - this is your domain.
- Identify the output values - this is your range.
- 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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### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Ex 1: Find Domain and Range of Ordered Pairs, Function or Not.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution. - College Algebra.
**Provided by:**OpenStax**Located at:**https://cnx.org/contents/[email protected]:1/Preface..**License:**CC BY: Attribution. - Ex: Give the Domain and Range Given the Points in a Table.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution. - Ex: Determine if a Table of Values Represents a Function.
**Authored by:**James Sousa (Mathispower4u.com) .**License:**CC BY: Attribution.