1. Copyright © Cengage Learning. All rights reserved. Piecewise Functions SECTION 7.2

2. 2 Learning Objectives 1 Define piecewise functions using equations, tables, graphs, and words 2 Determine function values of piecewise functions from a graph, equation, and table

3. 3 Piecewise Functions

4. 4 Consider Table 7.6, which shows the fees to park in the East Economy Garage at Sky Harbor International Airport in Phoenix, Arizona for a single day. We see that for any time over 0 minutes through 60 minutes, the fee is $4.00; for time over 60 through 120 minutes, the fee is $8.00; and for any time over 120 minutes (for one day), the fee is $10.00. Table 7.6

5. 5 Piecewise Functions Graphing this information as in Figure 7.13, we see that F(m) is a discontinuous combination of three linear functions. Figure 7.13 Sky Harbor International Airport Parking Fees

6. 6 Piecewise Functions Note that we use an open circle to denote that a value is not included in the function and an arrow to denote that the function continues beyond the graph. Figure 7.13 Sky Harbor International Airport Parking Fees

7. 7 Piecewise Functions A combination of functions such as the parking fee function is called a piecewise function.

8. 8 Defining a Piecewise Function with an Equation

9. 9 To model the parking-fee schedule using an algebraic equation, let’s look again at the different pieces that define F(m). For example, for any time up to and including 60 minutes, the parking fee is $4.00. We write Combining the separate equations for each level of parking fees, we have It is important to note that F(m) is a single function defined in many pieces, not many functions.

10. 10 Example 3 – Creating and Using a Piecewise Function Simply Fresh Designs is a small business that designs and prints creative greeting cards featuring client photos. The 2006 pricing structure for ordering personalized 4-inch by 6-inch holiday greeting cards from Simply Fresh Designs is shown in Table 7.7. Table 7.7

11. 11 Example 3 – Creating and Using a Piecewise Function a. Create a piecewise function that can be used to calculate the total cost of purchasing n cards. b. Use the piecewise function to calculate the cost of purchasing 40 cards and 50 cards. c. How many cards could we buy if we want to spend at most $45.00? cont’d

12. 12 Example 3(a) – Solution We let n represent the number of cards and C(n) represent the total cost (in dollars) of purchasing n cards. To determine the function conditions, we note that the card pricing changes from $1.00 per card to $0.75 per card when the order size reaches 50 cards. We also note the minimum order size is 25 cards. We have

13. 13 Example 3(a) – Solution To determine the function rules, we note that the total cost of the cards is a function of the individual card price plus the shipping cost ($6.00). If fewer than 50 cards are ordered the cost is. If 50 or more cards are ordered, the cost is. We now have cont’d

14. 14 Example 3(b) – Solution We are asked to use the piecewise function to calculate the cost of purchasing 40 cards and 50 cards. Due to the volume discount, the cost of 50 cards is actually $2.50 less than the 40-card cost. cont’d

15. 15 Example 3(c) – Solution To determine how many cards we could buy if we wanted to spend at most $45.00, we first use the first rule of the function. Since, the solution meets the required condition for the rule. We could order 39 cards for $45.00. cont’d

16. 16 Example 3(c) – Solution Now we consider the second rule of the function. Since 50  52, the solution meets the required condition for the rule. We could order 52 cards for $45.00. For a price of $45.00, we could either order 39 cards or 52 cards. In this case, we are able to get an additional 13 cards for no additional cost by understanding piecewise functions. cont’d

17. 17 Defining a Piecewise Function with an Equation When a scatter plot of a data set appears to have distinct pieces, we can use regression to find a model equation for each piece. The resulting piecewise function is often the best fit to the data.

18. 18 Example 4 – Using Regression to Create a Piecewise Function From 1981 to 1995, the annual number of adult and adolescent AIDS deaths in the United States increased dramatically. However, from 1995 to 2001, the annual death rate plummeted, as shown in Table 7.8. a. Draw a scatter plot of the data. b. Based on the scatter plot, what types of functions will best model each piece? Explain. c. Use regression to find a piecewise function, D(t), to model the AIDS death data. Table 7.8

19. 19 Example 4(a) – Solution The scatter plot is shown in Figure 7.14. Figure 7.14 Adult and Adolescent AIDS Deaths

20. 20 Example 4(b) – Solution From 1981 to 1995, the data is increasing. From 1995 until 2001, the data is decreasing. From 1981 until about 1989, the data appear concave up. A quadratic function may fit this piece well. Between 1989 and 1994, the data appear somewhat linear. A linear function may fit this piece well. Between 1995 and 2001, the scatter plot looks somewhat like a cubic function. cont’d

21. 21 Example 4(c) – Solution We use the data for each piece to calculate the corresponding regression models. For the quadratic model, we used data from t = 1 through t = 9. For the linear model, we used data from t = 9 through t = 14. For the cubic model, we used data from t = 15 through t = 21. The resultant function is cont’d

22. 22 Example 5 – Graphing a Piecewise Function from an Equation In 1998, Sammy Sosa of the Chicago Cubs and Mark McGuire of the St. Louis Cardinals were engaged in a race to break the all-time Major League Baseball single-season home run record set in 1962 by Roger Maris of the New York Yankees. As is shown in Figure 7.15, early in the season Sosa was hitting home runs at a relatively slow pace but in June he hit a record number of home runs. Figure 7.15

23. 23 Example 5 – Graphing a Piecewise Function from an Equation In the latter part of the season, his home run pace slowed from his scorching June pace but still remained quite brisk. The number of home runs Sosa hit in 1998 can be modeled by the piecewise function where H is the cumulative number of home runs hit and g is the number of games played. cont’d

24. 24 Example 5 – Graphing a Piecewise Function from an Equation a. Sketch a graph of H(g). Be sure to label the axes appropriately. cont’d

25. 25 Example 5(a) – Solution The graph is shown in Figure 7.16. Figure 7.16

26. 26 Example 5 – Graphing a Piecewise Function from an Equation b. Use the piecewise function of H(g) to evaluate H(30) and H(60). Explain what each solution means in its real-world context. cont’d

27. 27 Example 5(b) – Solution To evaluate H(30) means to find the cumulative number of home runs Sosa had hit after 30 games. Since g = 30 satisfies the condition 0  g < 55, we use the corresponding rule. Therefore, We estimate Sosa had hit 6 home runs after 30 games. cont’d

28. 28 Example 5(b) – Solution To evaluate H(60) means to find the number of home runs Sosa had hit after 60 games. Since g = 60 satisfies the condition 55  g  82, we use the corresponding rule. Therefore, We estimate Sosa had hit 18 home runs after 60 games. cont’d

29. 29 Example 5 – Graphing a Piecewise Function from an Equation c. Solve H(g) = 30 for g. cont’d

30. 30 Example 5(c) – Solution To solve H(g) = 30 for g, we need to use the graph in Figure 7.17 to find the game in which Sosa hit his 30th home run. Figure 7.17 cont’d

31. 31 Example 5(c) – Solution We locate 30 home runs on the vertical axis and then go over to the graph and then down to the horizontal axis to find the game that is associated with 30 home runs. It appears that at about game 75 Sosa hit his 30th homerun of the season. cont’d

32. 32 Example 5 – Graphing a Piecewise Function from an Equation d. Use H(g) to predict the number of home runs Sosa hit in 1998. Does the mathematical model show he would break Maris’s record of 62 home runs? Explain. (Note: The 1998 Major League Baseball season was 162 games, but the Cubs played 163 games in that season due to a one-game playoff with the San Francisco Giants.) cont’d

33. 33 Example 5(d) – Solution Since the Cubs played 163 games in 1998, we need to evaluate H(163). Since g = 163 satisfies the condition 82  g  163, we apply the corresponding rule. According to our model, Sosa hit 68 home runs in 1998 and broke Maris’s record of 62. (Actually, he hit 66 and broke the record set by Maris.) cont’d