1. 8-22-13 7-1,7-2 review A personal approach Math 94

2. Warm Up Describe the domain of each. Use appropriate notation.

3. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find each function value for the function f given by a) f(  3)b) f(2)c) f(7) Solution a) FOR f(-3) use f(x) = x + 3: f(  3) =  3 + 3 = 0 b) FOR f(2) use f(x) = x 2 ; f(2) = 2 2 = 4 c) FOR f(7) use f(x) = 4x = 4(7) = 28 Example

4. Eaxmple Think of three married couples you know. If you cannot think of any make them up. Write them down. Chris and Melanie, Dugg and Diana, Brad and Kathi Now write three ordered pairs describing the relationships. (Chris, Melanie) (Dugg, Diana) (Brad, Kathi)

5. Relation This is an example of a relation. (Chris, Melanie) (Dugg, Diana) (Brad, Kathi) Another relation is (1,2), (3,4), (5,6) The entries are called “coordinates”. 1 is the first coordinate, 2 is the second. Chris is the first coordinate, Melanie is the second.

6. Slide 7- 6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Relation A relation is -a set of ordered pairs. -a correspondence between a first set called the domain, and a second set, called the range. Note that for each member of the domain there is at least one member of the range. Think-If I only have one person, is that a relationship?

7. Domain and Range Notice the way my ordered pairs are written all the husbands are on the left and the wives are on the right. (Chris, Melanie) (Dugg, Diana) (Brad, Kathi) The husbands are the domain and the wives are the range. It is a relation because for EVERY husband there is a wife.

8. Domain and Range In this example (1,2), (2,3), (3,4), (4,5) my domain is {1,2,3,4} and my range is {2,3,4,5} Remember ordered pairs come in (x, y) form s the ones on left are x’s and the ones on the right are y’s. This is why we can say the domain is “all the x’s” and the range is “all the y’s”

9. Other ways to write relations Dugg Correspondence = Married To Chris Brad Kathi Diana Melanie Range Domain

10. Other ways to write relations Dugg Chris Brad Kathi Diana Melaniee

11. Other ways to write relations Dugg Chris Kathi Diana Melanie Brad x y

12. Independent Variable, Dependent Variable Now back to my relation (Chris, Melanie) (Dugg, Diana) (Brad, Kathi) Think who really depends on who. The wife depends on the husband for security and being taken care of. So the second coordinate depends on the first.

13. Dependable Needy

14. Which depends on which? The idea of dependence is what functions are about.

15. Slide 7- 15 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function A function is -a dependence relation. -A relation where y depends on x written y(x). -Since it is a function we replace y with f and write f(x).

16. More on functions A relation where for any member of the domain, there is exactly one member of the range. -This is also stated as for every x there is only one y. Marriage is a good example because for each husband there is only one wife. (Chris, Melanie) (Dugg, Diana) (Brad, Kathi)

17. Unless You live in a place where polygamy is legal. (Mike, Lisa) (Mike, Sally) (Mike, Sue)

18. Slide 7- 18 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The following relation is presented in two forms, map form and table form. Determine if the correspondence is a function. 8 0 –5 2 17 Solution The correspondence is a function because each member of the domain corresponds to exactly one member of the range. Example xy 82 017 -517

19. Slide 7- 19 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine if the correspondence is a function.Write the relation in table form. Jackson Max Cade Football Wrestling Soccer Solution The correspondence is not a function because a member of the domain (Jackson) corresponds to more than one member of the range. Example

20. Slide 7- 20 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine whether the correspondence is a function. A set of rectangles Range Solution The correspondence is a function, because each rectangle has only one area. Each rectangle’s area A set of numbers DomainCorrespondence Example

21. Slide 7- 21 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine if the correspondence is a function. Famous singers Range Solution The correspondence is not a function, because some singers have recorded more than one song. A song that the singer has recorded A set of song titles DomainCorrespondence Example

22. Slide 7- 22 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graph form If a function is a set of ordered pairs of numbers we can draw it in graph form. A function is in graph form if the ordered pairs are plotted. {(-5,1), (1,0), (4,3),(3,-5)} Why is this a function?

23. Slide 7- 23 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain and range of the function f below. Example

24. Slide 7- 24 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Here f can be written {(–5, 1), (1, 0), (3, –5), (4, 3)}. The domain is the set of all first coordinates, {–5, 1, 3, 4}, and the range is the set of all second coordinates, {1, 0, –5, 3}.

25. Slide 7- 25 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the coordinate It is common to ask which member of one set corresponds to a member of another. Return to my function. (Chris, Melanie) (Dugg, Diana) (Brad, Kathi). Which wife corresponds to Chris? Why? Note that this is like saying which member of the range corresponds to Chris. Chris is an x so we find the y that goes with Chris. Which member of the domain corresponds to Diana? Why is this a function?

26. {(-5,1), (1,0), (4,3),(3,-5)} Which member of the domain corresponds to 0? Which member of the range corresponds to -5?

27. Slide 7- 27 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For the continuous function f represented below, determine each of the following. a) What member of the range is paired with -2 b) What member of the domain is paired with 4 c) An x value for which f(x) = 3 y x -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 7 Example

28. Slide 7- 28 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley a) What member of the range is paired with -2 Solution The question is saying what member of the range is paired with -2 which means what y value corresponds to x = -2. So you are looking for a y value. Find x = -2 on the horizontal axis and go to the graph. The y- coordinate of the point is 3. Therefore 3 is the member of the range paired with -2. x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 Input Output 7

29. Slide 7- 29 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley b) What member of the domain is paired with 4 Solution The question is saying what member of the domain is paired with 4 which means what x value corresponds to y = 4. So you are looking for a x value. Find y = 4 on the graph and go to the x axis. The x-coordinate of the point is 1. Therefore 1 is the member of the domain paired with 4. x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 Input Output 7 How is this graph different from the previous example?

30. Slide 2- 30 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Vertical-Line Test This is a test to see if a graph is a function. It is more often used on continuous graphs. If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function. When a vertical line intersects more than once it represents multiple inputs with the same output.

31. Slide 2- 31 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Recall a graph is a set of ordered pairs. So graphs that do not represent functions are still relations. A function. Every vertical line intersects at most once. Not a function. Two y-values correspond to one x-value Not a function. Three y-values correspond to one x-value

32. Slide 7- 32 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Functions and Dependence One example of a function is a soda machine. The sodas (outputs) depend on the money (inputs). Can you think of another example of a function?

33. Slide 7- 33 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A nice visual The function pictured has been named f. Here x is an input, and f (x) – read “f of x,” is the corresponding output. With this notation y = f (x).

34. Slide 7- 34 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation and Equations In math most functions are described by equations. For example, f (x) = 5x +2 describes the function that takes an input x, multiplies it by 5 and then adds 2. f (x) = 5x + 2 To calculate the output f (3), take the input 3, multiply it by 5, and add 2 to get 17. That is, substitute 3 into the formula for f (x). Input f (3) = 5(3) + 2 = 17 Output

35. When I study my learning depends on my effort. Thus learning is a function of effort or L = f(e). y depends on x so y = f(x).

36. Slide 7- 36 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Find each indicated function value. a) f (–2), for f (x) = 3x 2 + 2x b) g(4), for g(t) = 6t + 9 c) h(m +2), for h(x) = 8x + 1 a) f (–2) = 3(–2) 2 + 2(–2) = 12 – 4 = 8 b) g(4) = 6(4) + 9 = 24 + 9 = 33 c) h(m +2) = 8(m+ 2) + 1 = 8m + 16 + 1 = 8m + 17 Example

37. Slide 7- 37 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Note that whether we write f (x) = 5x +2 or f (m) = 5m +2, we still have f (3) = 17. Thus the independent variable can be thought of as a dummy variable. When a function is described by an equation, the domain is often unspecified. In such cases, the domain is the set of all numbers for which function values can be calculated.

38. Slide 7- 38 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley A small business started out in the year 1996 with 10 employees. By the start of 2000 there were 28 employees, and by the beginning of 2004 the business had grown to 34 employees. Estimate the number of employees in 1998 and also predict the number of employees in 2007. Example

39. Slide 7- 39 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Write the relation in ordered pair form and graph form. Use the graph to answer the question. Plot the points and connect the three points. Let the horizontal axis represent the year and the vertical axis the number of employees. Label the function itself E. 10 30 20 40 50 200420001996 Number of Employees Year

40. Slide 7- 40 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. Using the graph. To estimate the number of employees in 1998, locate the point directly above the year 1998. Then estimate its second coordinate by moving horizontally from that point to the y-axis. We see that 10 30 20 40 50 200420001996 19 1998 Year Number of Employees

41. Slide 7- 41 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3. Using the graph (continued). To predict the number of employees in 2007, extend the graph and extrapolate. We see that 10 30 20 40 50 2004200019962007 Year Number of Employees

42. Slide 7- 42 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4. Check. A precise check would involve knowing more information. Since 19 is between 10 and 28 and 40 is greater than 34, the estimate seems plausible. 5. State. In 1997, there were about 19 employees at the small business. By 2007, the number of employees should grow to 40.

43. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domain and Range Determining the Domain and Range Restrictions on Domain Functions Defined Piecewise 7.2

44. Slide 7- 44 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When a function is in ordered pair form, the domain is the set of all first coordinates and the range is the set of all second coordinates. Find the domain and range for the function f given by f = {(–5, 1), (1, 0), (3, –5), (4, 3)}. Solution The domain is the set of all first coordinates: {–5, 1, 3, 4}. The range is the set of all second coordinates: {1, 0, –5, 3}. Example

45. Slide 7- 45 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain and range of the function f in continuous graph from. y x -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 7 Example

46. Slide 7- 46 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution x y -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 The domain of f 7 The domain of f is the set of all x-values of the points on the curve. These extend continuously from -5 to 3 and can be viewed as the curve’s shadow, or projection, on the x-axis. Thus the domain in set interval notation is the domain is the set of all first coordinates and the range is the set of all second coordinates.

47. Slide 7- 47 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The range of f Solution The range of f is the set of all y-values of the points on the curve. These extend continuously from -1 to 7 and can be viewed as the curve’s shadow, or projection, on the y-axis. Thus the range in set interval notation is the domain is the set of all first coordinates and the range is the set of all second coordinates.

48. Slide 7- 48 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Recall When a function is described by an equation, the domain is often unspecified. In such cases, the domain is the set of all numbers for which function values can be calculated.

49. Slide 7- 49 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine the domain of Solution We ask, “Is there any number x for which we cannot compute 3x 2 – 4?” Since the answer is no, the domain of f is the set of all real numbers. Example

50. Slide 7- 50 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine the domain of Solution We ask, “Is there any number x for which cannot be computed?” Since cannot be computed when x – 8 = 0 the answer is yes. x – 8 = 0, x = 8 Thus 8 is not in the domain of f, whereas all other real numbers are. The domain of f is To determine what x-value would cause x – 8 to be 0, we solve an equation: Example

51. Slide 7- 51 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Functions Defined Piecewise A piecewise function is a function whose equation differs according its domain. These functions are piecewise defined. To find f(x) for a piecewise function a)Determine what part of the domain x belongs to. b)Then use the equation for that part of the domain.

52. IC

53. Quiz