1. 4-3 Relations Objectives Students will be able to: 1) Represent relations as sets of ordered pairs, tables, mappings, and graphs 2) Find the inverse of a relation

2. Terminology Relation: set of ordered pairs Domain: set of all x values in a relation Range: set of all y values in a relation

3. Ways to Represent a Relation 1)As a set of ordered pairs Example: {(1, 2), (-2, 4), (0, -3)} 2) As a table 3) As a graph

4. Example 1: Express each relation as a table, a graph, and a mapping. Then determine the domain and range. a){(4, 3), (-2, -1), (-3, 3), (2, -4)} Domain:Range:

5. You try. b) {(3, 2), (5, 2), (3, -1), (0, 1)} Domain:Range:

6. Inverse: relation obtained from switching the coordinates of each ordered pair of the original relation For example, if a relation is {(2, 1), (3, -5), (0,1)}, its inverse would be {(1, 2), (-5, 3), (1, 0)}. Try and find the inverse of this relation. List the inverse as a set of ordered pairs.

7. 4-6 Functions Objectives Students will be able to: 1)Determine whether a relation is a function 2)Find functional values Note: You cannot spell function without “fun”

8. Functions A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.

9. Let’s talk about what this means by looking at a real-life example of a relation. Let’s say that our domain is students, and our range is television shows. We can create a mapping of the relation.

10. Let’s now recap using mathematical examples:

11. Example 2: Determine if the relation is a function. a){(1, 3), (2, 3), (-1, 1)} yes b) {(1, 4), (2, 1), (1, 5)} No; the x value of 1 repeats Try c) {(3, 1), (3, 2), (3, 4)}d) {(1, -1), (2, -1)} No yes

12. Vertical Line Test When given a graph of a relation, one can perform a vertical line test to determine whether a relation is a function. If you drop in vertical lines, and they do not intersect the graph in more than one point, then the relation is a function. If they do intersect the graph in more than one point, then the relation is not a function.

13. Vertical Line Test

14. Example 3: Use the vertical line test to determine if the relation is a function. a)b) Not a function Yes, is a function

15. c)d) yes no

16. Try these: e)f) Not a function Yes, is a function

17. A further look at domain and range Remember that a domain is the set of our x values, and a range is the set of our y values. We can also determine the domain and range for linear equations, quadratic equations, absolute value equations, all types of equations.

18. Example 4: For each graph, determine the domain and range. a) Domain: Range: b) Domain: Range:

19. c)Domain: Range: d)Domain: Range:

20. Try these. e) f)Domain:Range:

21. Function Notation

22. a)f(-2)b) g(5)c) f(2d) Try these. d) g(-4)e) f(3p)f) g(2a)