1. Math 104 1-14-14 Intro to functions

2. I am Mr. Fioritto. You are Math 104 Spring 14. We meet from 9:30-11:45 on T, Th We will use Intermediate Algebra 9 th ed. By Bittinger,Ellenbogen,Johnson We will use my math lab but it will not be mandatory. Email is cfioritto@napavalley.edu or c_fioritto@yahoo.comcfioritto@napavalley.edu c_fioritto@yahoo.com Phone is 707-673-7592 Office Hour is Thursday 9:00 – 9:30 in the classroom. Don’t miss class. Four absences will result in a drop. Don’t cheat. Cheating results in a zero when caught. No cell phones in class. Check out the math lab. Our class website is www.cfioritto.wordpress.com Homework is given daily upon section completion and due on Thursday. Everyday starts with a warm up, goes to instruction, goes to break, goes to IC, finishes with quiz. We test at the end of every chapter. We will take a cumulative final. My grading scale is 20,20,20,40. Hw, Ic, and Quizzes are all 20% while tests are 40%. Questions? Lets get started. Important Information

3. a) Plot the points (2,4) (3,5) (0, 3) (-5,0) In any set of ordered pairs, the numbers are called x and y coordinates. x is always first and y is always second. x coordinates are sometimes called inputs and y coordinates are called outputs or “values”. b) What is the value above when the input is 3? c) What inputs,if any, above give a value of 0? Warm Up

4. Students will -Determine if a relation is a function. -Find the domain and range of a function. -Find function values. -Draw functions in table form, map form, graph form. Objectives

5. Slide 7- 5 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Relation A relation is a set of ordered pairs. Give an example of an ordered pair. What is the x-value? The y-value? Definition

6. Slide 7- 6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley These are all relations. What are the domains and ranges of each relation? -A set of students and their textbooks. -A set of textbooks and their pages. -A set of presidents and their terms. -(1,2), (2,3), (3,4), (4,5) -y = x+1 Example

7. What is a relation? Give an example of a relation. Why is y = x + 1 a relation? Some people say a relation is a correspondence. Why? Your Turn

8. Slide 7- 8 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function A relation where for every x there is only one y. Is every relation a function? Why or why not? Is every function a relation? Why or why not? Definition

9. Slide 7- 9 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Which relations are functions? - A set of lines and their slopes. -A set of numbers and their squares. -y = x+1 (1,2), (2,3), (1,4), (4,5) State the definition of a function. Example

10. -A set of numbers and their square roots. -A set of presidents and their terms. -A set of football players and their superbowl rings. -(1,2), (2,3), (3,4), (4,5) Which relations are functions? Your Turn What is a function? Give an example of a function.

11. Forms of Functions There are four forms of functions Map Table Ordered Pair Graph Definition

12. Slide 7- 12 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The relation (8,2), (0,17), and (-5,17) is presented below in two forms, map form and table form. 8 0 –5 2 17 Example xy 82 017 -517 Map form and Table form

13. Slide 7- 13 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ordered pair form and Graph form A function is in ordered pair form if it is a list of ordered pair like A function is in graph form if all of its ordered pairs are plotted on a graph. Why is this a function? {(-5,1), (1,0), (4,3),(3,-5)} Example

14. 8 0 –5 2 17 {(-5,1), (1,0), (4,3),(3,-5)} xy 82 017 -517 For each relation state the form it is in and draw it in the other three forms. Your Turn

15. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Domain and Range Definition In any function or relation, the domain is the set of all x coordinates and the range is the set of all y coordinates.

16. Slide 7- 16 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain and range for the function f given by f = {(–5, 1), (1, 0), (3, –5), (4, 3)}. Example

17. Your Turn Find the domain and range for the relation given by f = {(–7, 0), (3, 4), (3, –2), (4, 5)}.

18. Slide 7- 18 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain of the function f below. Example

19. Slide 7- 19 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} Example continued

20. The range is the set of all second coordinates, {1, 0, –5, 3}. Find the range of the function f below. Your Turn

21. Definition Domains and ranges of continuous functions are always intervals.

22. Slide 7- 22 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain and range of the function f. y x -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 7 Example

23. Slide 7- 23 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 in set interval notation is the domain is the set of all first coordinates and the range is the set of all second coordinates. Example

24. Slide 7- 24 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The range of f Solution 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. Example

25. Find the domain and range of the function f. Your Turn

26. Slide 7- 26 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When a function is described by an equation like, the domain is often unspecified. In such cases, the domain is the set of all numbers for which function values can be calculated. Definition

27. Slide 7- 27 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

28. Slide 7- 28 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: Your Turn

29. Definition Values of function The value of a function at a given point is the y coordinate of that point.

30. Slide 7- 30 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For the 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 = f(x) x -5 -4 -3 -2 -1 1 2 3 4 5 f 4 -2 -4 -3 3 2 5 1 6 7 Example

31. a) What member of the range is paired with 0? b) What member of the domain is paired with 3? c) An x value for which f(x) = 0. Your Turn

32. Definition To relate these concepts to real world applications it is essential to draw a graph.

33. Slide 7- 33 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

34. Slide 7- 34 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 Example

35. Slide 7- 35 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 Example

36. Slide 7- 36 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 Example

37. Slide 7- 37 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. Example

38. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Vertical-Line Test 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. Definition

39. Use the vertical line test to decide which graphs above are functions. Example

40. Slide 2- 40 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Your Turn Use the vertical line test to decide which graphs above are functions.

41. Slide 7- 41 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation and Equations To understand function notation f(x), it helps to imagine a “function machine.” Think of putting a member of the domain (like quarters) into the machine. The machine is programmed to produce the appropriate member of the range (like sodas) Definition

42. Slide 7- 42 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation and Equations 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). Example

43. Slide 7- 43 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Function Notation and Equations Most functions can be 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 Example

44. Slide 7- 44 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 Your Turn

45. Slide 7- 45 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. Definition

46. 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

47. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find each function value for the function f given by a) f(  4) b) f(4) c) f(0) Your Turn

48. Quiz Find the domain of each function.