1. Thursday

4. Relations and Functions Chapter 2 Section 2-1 Pages 72-81

5. Objectives I can find Domain and Range I can determine if the relation is a function by two methods IMPORTANT VOCABULARY in this section!!

6. Important Vocabulary Relation Domain Range Function Vertical Line Test

7. Cartesian Coordinate Plot x-axis: This is the horizontal axis. y-axis: This is the vertical axis Origin: This is the center point (0,0) Each point on the coordinate plane can be represented by an ordered pair (x,y), where x is the distance from Origin on the X-Axis and y is the distance from Origin on the Y-Axis.

8. Quadrants The coordinate system is divided into 4 quadrants. Look at the overhead slide 2-1.1 to see how this entire system works.

9. Quadrant I (+, +) Quadrant II (-, +) Quadrant IV (+, -) Quadrant III (-, -) Origin (0,0)

10. Relation A relation is a set of ordered pairs! Need the { } to show a set Example: { (1, 2), (3, 4), (5, 6) }

11. Domain and Range The domain in any relation is the first coordinates from the ordered pairs. It is the Input! Domain = X -Values The range in any relation is the second coordinates from the ordered pairs. It is the Output! Range = Y- Values

12. Dependent Variable

13. Example 1: Domain/Range Given the following relation {(2,3), (-4,8), (2,6), (7,-3)} What is the Domain? { -4, 2, 7} **Notice they are listed least to greatest!! No duplicates!!! What is the Range? {-3, 3, 6, 8}

14. Example 2: Given the following ordered pairs, find the domain and range. {(4,5), (-2,3), (5,6)} Domain is {-2, 4, 5} Range is {3, 5, 6}

15. Answer Format When listing a set of numbers for domain or range, use the set symbols {} List numbers from least to greatest (increasing order). No duplicates! Ex: the domain has numbers: 3, -2, 5, 2, 3 {-2, 2, 3, 5}

16. SOLUTION GUIDED PRACTICE for Example 1 1. Identify the domain and range of the function. Input 0124 Output 5221 The domain is the set of inputs: {0,1,2,4} The range is the set of outputs: {1,2,5}

17. 4 Ways to see Relations

18. Relations Ordered Pairs (2, 3) (-3, 1) (1, -2) XY 23 -31 1-2 Tables Graphs Mapping 2 -3 1 3 1 -2 XY

19. Function A function is a special relation in which NO DUPLICATED “x-values” Example: Is the following relation a function: { (1,3), (4,-9), (6,3) } Answer: Yes. Each element from the domain has only 1 element from the range

20. Ex 2: How about this relation. Is it a function? Given the following { (2,3), (-4,8), (2,6), (7,-3)} Domain? { -4, 2, 7} Range? {-3, 3, 6, 8} Function: No. the element 2 from the domain is mapped to 2 elements in the range.

21. Tell whether the pairing is a function. Identify a function EXAMPLE 2 a. a. The pairing is not a function because the input 0 is paired with both 2 and 3.

22. b.b. Identify a function EXAMPLE 2 Output Input 21 00 4 8 612 The pairing is a function because each input is paired with exactly one output.

23. SOLUTION GUIDED PRACTICE for Example 2 Tell whether the pairing is a function. 1221 Output 12963 Input 2. 3 6 9 12 12211221 The pairing is a function because each input is paired with exactly one output.

24. SOLUTION GUIDED PRACTICE for Example 2 Tell whether the pairing is a function. 3210 Output 7422 Input 3. The pairing is not a function because each input is not paired with exactly one output. 247247 01230123

25. Vertical Line Test You can use a vertical line test to easily see on a graph is the relation is a function. You place a straight edge like a pencil vertical on the graph and move it across the graph. If one line intersects the graph at only one point, then it is a function.

26. Applying VLT

27. y 2 = x x y Vertical Line Test Consider the graphs. x 2 + y 2 = 1 x y y = x 2 x y 2 points of intersection 1 point of intersection 2 points of intersection Vertical Line Test

28. Examples VLT

29. VLT Applies to Grid

30. ½ Sheet Activity On your ½ sheet of paper Make 4 relations that we discussed today. ONE of each type Trade with your partner and detrmine –Domain and Range –Is each a function?

31. Homework WS 1-5