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

3. Relations A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain . The set of output values is called the range .

4. Domain & Range Domain is the set of all x values. Range is the set of all y values. Example 1: Domain- D: {1, 2} Range- R: {1, 2, 3} {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}

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

8. Cartesian Coordinate System Cartesian coordinate plane x-axis y-axis origin quadrants

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?

16. Functions A relation as a function provided there is exactly one output for each input. It is NOT a function if at least one input has more than one output

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

18. Example 6 No two ordered pairs can have the same first coordinate (and different second coordinates). Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}

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)}

22. The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function.

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

25. Examples I’m going to show you a series of graphs. Determine whether or not these graphs are functions. You do not need to draw the graphs in your notes.

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?