1. PRE-ALGEBRA

2. Lesson 8-1 Warm-Up

3. PRE-ALGEBRA Relations and Functions (8-1) What is a relation? What is the “domain” of a relation? What is the “range” of a relation? What is a mapping diagram? relation: a set of ordered pairs Example: The (age, height) ordered pairs below form a relation. You can list the set of ordered pairs in a relation using braces --- {} {(18, 4.0), (16, 4.5), (20, 5.5), 14, 5.0)} domain: the set (group) of all of the x-coordinates of a relation listed in order from least to greatest Example: The domain of the above relation is (14, 16, 18, 20) range: the set (group) of all of the y-coordinates of a relation listed in order from least to greatest Example: The range of the above relation is (4.0, 4.5, 5.0, 5.5) mapping diagram: a diagram in which the domain values are matched with their range values using arrows Example: The following is a mapping diagram of the above giraffe relation.

4. PRE-ALGEBRA Relations and Functions (8-1) What is a function? function: a relation in which each value in the range matches up with exactly one value in the range (in other words, there aren’t two or more coordinates with the same y-coordinate). Examples: The mapping diagram on the left is a function, because no value in the domain corresponds (matches up to), more than one range value. The mapping diagram on the right is NOT a function, because there are two range values for the domain value of 1.

5. PRE-ALGEBRA Is each relation a function? Explain. a. {(0, 5), (1, 6), (2, 4), (3, 7)} List the domain values and the range values in order. Draw arrows from the domain values to their range values. There is one range value for each domain value. This relation is a function. Relations and Functions LESSON 8-1 Additional Examples

6. PRE-ALGEBRA (continued) b. {(0, 5), (1, 5), (2, 6), (3, 7)} There is one range value for each domain value. This relation is a function. Relations and Functions LESSON 8-1 Additional Examples

7. PRE-ALGEBRA (continued) c. {(0, 5), (0, 6), (1, 6), (2, 7)} There are two range values for the domain value 0. This relation is not a function. Relations and Functions LESSON 8-1 Additional Examples

8. PRE-ALGEBRA Relations and Functions (8-1) How do you create a mapping diagram? What is the vertical line test? To create a mapping diagram, list the domain and range values in order. Then, draw arrows from the domain values to their range values. Example: Is the relation [(0,1), (1,2), (1,3), (2,4)} a function? List the domain and range values in order Draw arrows from the domain values to their range values. The relation is a function, because there is only one range value for each domain value. Vertical-Line Test: On a graph, if a vertical (up-down) line passes through more than one coordinate, that means that at least one x- coordinate shares more than one y-coordinate, which means the relation is not a function (in other words, a relation that doesn’t pass the vertical line test is not a function).

9. PRE-ALGEBRA Relations and Functions (8-1) How do you use the vertical line test? Example: The following relation is not a function, because a vertical line (like a pencil) passes through (3,0) and (3,2). To use the vertical line test, graph the function to determine whether a vertical line will pass through two or more points. Example: Use the vertical line test to determine if the following table is a function. 1. Graph the function. 2. Determine if a vertical line will pass through two or more like.. The relation is not a function, because a vertical line would pass through (2,0 and (2, 3).

10. PRE-ALGEBRA Is the time needed to mow a lawn a function of the size of the lawn? Explain. No; two lawns of the same size (domain value) can require different lengths of time (range values) for mowing. Relations and Functions LESSON 8-1 Additional Examples

11. PRE-ALGEBRA a. Graph the relation shown in the table. Domain Value –3 –5 3 5 Range Value 5 3 5 3 Relations and Functions LESSON 8-1 Additional Examples Graph the ordered pairs (–3, 5), (–5, 3), (3, 5), and (5, 3).

12. PRE-ALGEBRA b. Use the vertical-line test. Is the relation a function? Explain. (continued) The pencil does not pass through two points at any one of its positions, so the relation is a function. Pass a pencil across the graph as shown. Keep the pencil vertical (parallel to the y-axis) to represent a vertical line. Relations and Functions LESSON 8-1 Additional Examples

13. PRE-ALGEBRA Is each relation a function? Explain. 1.{(–2, –1), (4, 2), (–8, –4), (6, 3)} 2.{(5, 0), (7, 2), (9, 4), (5, 1)} 3.Graph the relation in the table. Is the relation a function? Explain. xy –17 07 17 27 Yes; there is only one range value for each domain value. No; the domain value 5 has two range values, 0 and 1. Yes; there is one range value for each domain value. Check students’ graphs. Relations and Functions LESSON 8-1 Lesson Quiz