Graphs and Functions
Relations and Functions
Learning Objectives
By the end of this section, you will be able to:
- Find the domain and range of a relation
- Determine if a relation is a function
- Find the value of a function
Before you get started, take this readiness quiz.
Find the Domain and Range of a Relation
As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.
When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation.
There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.
A relation is any set of ordered pairs, All the x-values in the ordered pairs together make up the domain. All the y-values in the ordered pairs together make up the range.
For the relation
ⓐ Find the domain of the relation.
ⓑ Find the range of the relation.
ⓐ The domain is the set of all x-values of the relation.
ⓑ The range is the set of all y-values of the relation.
For the relation
ⓐ Find the domain of the relation.
ⓑ Find the range of the relation.
ⓐ
ⓑ
For the relation
ⓐ Find the domain of the relation.
ⓑ Find the range of the relation.
ⓐ
ⓑ
A mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.
Use the mapping of the relation shown to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
ⓐ The arrow shows the matching of the person to their birthday. We create ordered pairs with the person’s name as the x-value and their birthday as the y-value.
{(Alison, April 25), (Penelope, May 23), (June, August 2), (Gregory, September 15), (Geoffrey, January 12), (Lauren, May 10), (Stephen, July 24), (Alice, February 3), (Liz, August 2), (Danny, July 24)}
ⓑ The domain is the set of all x-values of the relation.
{Alison, Penelope, June, Gregory, Geoffrey, Lauren, Stephen, Alice, Liz, Danny}
ⓒ The range is the set of all y-values of the relation.
{January 12, February 3, April 25, May 10, May 23, July 24, August 2, September 15}
Use the mapping of the relation shown to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
ⓐ (Khanh Nguyen, kn68413), (Abigail Brown, ab56781), (Sumantha Mi