1. The document discusses relations and functions, including identifying their domain and range. It provides examples of plotting points on a Cartesian plane and using graphs to represent equations. 2. Functions are defined as relations where each input is mapped to only one output. Several examples are given to demonstrate determining if a relation qualifies as a function using techniques like the vertical line test. 3. The key concepts of domain, range, and using graphs are illustrated through multiple examples of sketching and analyzing relations and functions.

1. Warm Up Activity

2. Relation and Function Objective: 1. Identify Domain and Range 2. Use the Cartesian Plane in plotting points 3. Graph equations using a chart 4. Determine if a Relation is a Function 5. Use the Vertical Line Test for Functions

5. Example 2: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e} Range: {1, 2, 3}

6. Can you give example/s of relation you use or experience daily?

7. Graphs

9. A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X<0, y>0 Quadrant III X<0, y<0 Quadrant IV X>0, y<0 Origin (0,0)

10. Plot: (-3,5) (-4,-2) (4,3) (3,-4)

11. Every equation has solution points (points which satisfy the equation). 3x + y = 5 (0, 5), (1, 2), (2, -1), (3, -4) Some solution points: Most equations have infinitely many solution points.

12. Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4) and (7, 5); y = 3x -1 The collection of all solution points is the graph of the equation.

13. Ex4 . Graph y = 3x – 1. x 3x-1 y

14. Ex 5. Graph y = x ² - 5 x x ² - 5 y -3 -2 -1 0 1 2 3

15. What are your questions?

17. Functions INPUT (DOMAIN) OUTPUT (RANGE) FUNCTION MACHINE In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT

19. Identify the Domain and Range. Then tell if the relation is a function. Input Output -3 3 1 1 3 -2 4 Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output

20. Input Output -3 3 1 -2 4 1 4 Identify the Domain and Range. Then tell if the relation is a function. Domain = {-3, 1,4} Range = {3,-2,1,4} Function? No: input 1 is mapped onto Both -2 & 1 Notice the set notation!!!

21. Is this a function? 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)}

23. (-3,3) (4,4) (1,1) (1,-2) Use the vertical line test to visually check if the relation is a function. Function? No, Two points are on The same vertical line.

24. (-3,3) (4,-2) (1,1) (3,1) Use the vertical line test to visually check if the relation is a function. Function? Yes, no two points are on the same vertical line

26. #1 Function? YES!

27. Function? #2 YES!

28. Function? #3 NO!

29. Function? #4 YES!

30. Function? #5 NO!

31. #6 Function? YES!

32. Function? #7 NO!

33. Function? #8 NO!

34. #9 Function? YES!

35. Function? #10 YES!

36. Function? #11 NO!

37. Function? #12 YES!

38. Function Notation “ f of x” Input = x Output = f(x) = y

39. y = 6 – 3x -2 -1 0 1 2 12 9 6 0 3 f(x) = 6 – 3x -2 -1 0 1 2 12 9 6 0 3 Before… Now… (x, y) (input, output) (x, f(x)) x y x f(x)

40. Find g (2) and g (5). g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)} g(2) = 3 g(5) = 2 Example 7

41. Consider the function h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)} Example 8 Find h(9), h(6), and h(0).

42. Example 9 f(x) = 2x 2 – 3 Find f(0), f(-3), f(5a).

43. F(x) = 3x 2 +1 Example 10 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a 2 + 1

44. Domain The set of all real numbers that you can plug into the function. D: {-3, -1, 0, 2, 4}

45. What is the domain? g(x) = -3x 2 + 4x + 5 D: all real numbers Ex. Ex . x + 3  0 x  -3 D: All real numbers except -3

46. What is the domain? x - 5  0 Ex. D: All real numbers except 5 D: All Real Numbers except -2 Ex. x + 2  0 h x x ( )   1 5 f x x ( )   1 2

47. What are your questions?