3-1 Reteach to Build UnderstandingRelations and Functions1. The domain is the set of x -values and the range is the set of y -values. A relation is any set of ordered pairs. A relation is a function when each input, or element in the domain, has exactly one unique output, or element in the range.2. Pilar is given the following set of ordered pairs. (2,4),(4,6),(6,8),(8,10), (12,8). Read her statements. Pilar incorrectly identified two of the key features of relations and functions. Put an X next to any incorrect statements. Correct her errors.a. The domain is 2,4,6,8.b. The range is 4,6,8,10.c. The set of ordered pairs is a relation.d. The relation is a function because it passes the vertical line test.e. The relation is not a function because two inputs go to the same output.3. Identify the domain and range of each relation. Fill in the correct number(s) in the blanks to identify the domain and range. Then tell whether or not it is a function by circling the correct response.a. (2,3),(4,6),(1,5),(2,5),(0,5)Domain: 0, 1, 2,_Range: 3, 5,_b. (3,4),(5,4),(7,4),(8,4),(10,4)Domain: 3,5,7, ._Range: _

1. Image Analysis:
   a. Circle the correct domain and range:
   • Domain: \(\{ 2, 3, 4, 5, 6\} \)
   • Range: \(\{ 4, 6, 8, 10, 12\} \)
     b. Circle the relation that is a function:
   • First table (each x-value maps to one y-value).
2. Pilar's Statements:
   • Identify the domain and range from the given ordered pairs.
   • Domain: \(\{ 2, 4, 6, 8, 12\} \)
   • Range: \(\{ 4, 6, 8, 10\} \)
   • Check each statement:
     • a. Incorrect, missing 12.
     • b. Correct.
     • c. Correct.
     • d. Incorrect, fails vertical line test.
     • e. Correct, two inputs (6 and 12) have the same output (8).
3. Identify Domain and Range:
   a. Relation: \(( 2, 3) , ( 4, 6) , ( 1, 5) , ( 2, 5) , ( 0, 5) \)
   • Domain: \(\{ 0, 1, 2, 4\} \)
   • Range: \(\{ 3, 5, 6\} \)
   • Not a function (input 2 has multiple outputs).
     b. Relation: \(( 3, 4) , ( 5, 4) , ( 7, 4) , ( 8, 4) , ( 10, 4) \)
   • Domain: \(\{ 3, 5, 7, 8, 10\} \)
   • Range: \(\{ 4\} \)
   • Is a function (each input has one output).

Supplemental Knowledge

Understanding relations and functions are central concepts in algebra. A relation is any ordered pair; while functions are specific types of relations in which every element in their domain (set of x-values) maps to one unique value in their range (set of y-values). No two x-values may ever pair more than once with multiple y-values - thus no two consecutive x values may ever pair more than once with multiple y values. A visual way of testing whether a relation qualifies as a function exists is the vertical line test: If any vertical lines intersects multiple points then that indicatess this test resulted in conclusive evidence against its potential as being classified as not being classified as such a function.

Real-World Applications

Imagine an online store in which each product has a distinct ID (domain), with each ID matching exactly one price (range). When customers select an item by its ID number, they get only the one price associated with its ID; otherwise it would cause confusion for both customers and inventory systems alike.

Understanding relations and functions is crucial for solving many real-world problems efficiently. At UpStudy, we provide comprehensive resources to help you master these concepts. Our live tutor question bank and AI-powered problem-solving services can guide you through complex topics with ease.
For further practice on identifying domains and ranges or understanding functions better, explore UpStudy’s Algebra Functions Calculator! It’s designed to help you grasp these concepts effortlessly!