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Copyright © Cengage Learning. All rights reserved. Composition of Functions SECTION 7.3

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2 Learning Objectives 1 Compose two or more functions using tables, equations, or graphs 2 Create, use, and interpret function composition notation in a real-world context

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3 Understanding Composition of Functions

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4 Consider Table 7.10, which shows the 2004 median annual salaries of American employees, the associated educational degree acquired, and the accumulated full-time academic years typically spent to attain such degrees. Table 7.10

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5 Example 1 – Understanding Composition of Functions Use Table 7.10 to answer the following questions. a. If an employee spent 4 years in college, what degree did she earn?

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6 Example 1(a) – Solution According to Table 7.10, a person who attends college for 4 years earns a Bachelor’s degree. Notice academic years is the input of the function and degree is the output. The function used does not reveal anything about the person’s median annual salary. Table 7.10

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7 Example 1 – Understanding Composition of Functions Use Table 7.10 to answer the following questions. b. What is the median annual salary of an employee with a Doctorate?

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8 Example 1(b) – Solution According to the table, an employee with a Doctorate degree earns a median annual salary of $72,073. Observe degree attained is the input of the function and median annual salary is the output. This function does not reveal anything about the academic years spent in college. cont’d

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9 Example 1 – Understanding Composition of Functions Use Table 7.10 to answer the following questions. c. If an employee attended school for 6 years, what degree did he earn? What is the median annual salary of an employee with a Master’s degree?

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10 Example 1(c) – Solution According to the table, a person who attends college for 6 years earns a Master’s degree. The median annual salary of a person with a Master’s degree is $50,693. For the first question, academic years is the input and degree attained is the output. However, for the second question, the degree attained is the input and the median annual salary is the output of the function. cont’d

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11 Example 1 – Understanding Composition of Functions Use Table 7.10 to answer the following questions. d. What is the median annual salary of an employee who spent 6 years in college?

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12 Example 1(d) – Solution According to the table, an employee with 6 years of college earns a median annual salary of $50,693. (This assumes the employee earned a Master’s degree.) We are using a function that has academic years as the input and median annual salary as the output. In effect, we skipped the step of finding the employee’s degree and calculated the salary directly. cont’d

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13 Understanding Composition of Functions In parts (c) and (d) of Example 1 we arrived at the same answer for the employee’s median annual salary but took two different paths to get there. Table 7.11

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14 Understanding Composition of Functions The process outlined in part (d) demonstrates function composition, which occurs when the output of one function is used as an input to another function. In this case, the output of the degree attained function was the input to the median annual salary function.

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15 Composition Notation

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16 Composition Notation Continuing with the salary example, we define the variables as follows: t is the accumulated time in college (academic years). d is the degree attained. s is the median annual salary. It is important to understand the relationship between these variables. d and t We know d is a function of t because the degree attained by an employee depends on the accumulated number of academic years. We write the relationship as d(t).

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17 Composition Notation s and d We know s is a function of d because the median annual salary of an employee depends on the degree attained. We write the relationship as s(d). How are s, d, & t related?

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18 Composition Notation s, d, and t We have shown that s is a function of d and that d is a function of t. That is, salary (s) depends on the degree earned (d), which depends on the accumulated number of years spent in college (t). However, we have also seen these two functions can be combined in a composition of functions: s is a function of d and d is a function of t. This relationship is sometimes represented with the notation (s d)(t); however, we will use the more intuitive notation, s(d(t)).

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19 Composition Notation

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20 Creating Composite Functions

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21 Example 2 – Creating a Composite Function Defined by Tables More than 500 new energy drinks were launched worldwide in 2006, each one with promises of weight loss, increased endurance, and legal highs. Although energy drinks such as Red Bull, Rock Star, Rush, Monster, and Hype contain some nutritional vitamins and supplements, many contain other ingredients that are not necessarily good for you. One particular brand of energy drink has 80 milligrams of caffeine in an 8.3-ounce can. cont’d

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22 Example 2 – Creating a Composite Function Defined by Tables The effects of caffeine in the bloodstream on the heart rate of a typical 175-pound male adult are displayed in Table 7.12. It is also known that the amount of caffeine in a person’s bloodstream dissipates over time. Table 7.12 cont’d

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23 Example 2 – Creating a Composite Function Defined by Tables The amount of caffeine in a 175-pound adult male’s bloodstream would typically dissipate as shown in Table 7.13, where time, t, is the number of hours after the caffeine is ingested. cont’d Table 7.13

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24 Example 2 – Creating a Composite Function Defined by Tables Using Tables 7.12 and 7.13, create a table that shows the heart rate, r, as a function of the time, t. Solution: We begin by writing the two tables adjacent to each other, as shown in Tables 7.14 and 7.15. cont’d Table 7.14Table 7.15

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25 Example 2 – Solution We see that the values for the caffeine levels appear to be the same in many cases. We create a three-column table (Table 7.16) to capture what we know. cont’d Table 7.16

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26 Example 2 – Solution We now eliminate the rows of the table with blank spaces and delete the middle column to get Table 7.17, which relates time, t, to heart rate, r. So r(c(t)) is the composite function. cont’d Table 7.17

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27 Example 7 – Decomposing Functions Symbolically The given function is a composite function of the form f (g(x)). Decompose the functions into two functions f (x) and g(x). a.

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28 Example 7 – Solution a. The function f (g(x)) takes the absolute value of 3x – 2. Two functions that can be used for the composition are f (x) = |x| and g(x) = 3x – 2. This is not a unique solution; f (x) = |x – 2| and g(x) = 3x also work.

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29 Example 7 – Decomposing Functions Symbolically Each given function is a composite function of the form f (g(x)). Decompose the functions into two functions f (x) and g(x). b.

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30 Example 7 – Solution b. The function f (g(x)) consists of a rational function and a quadratic function. Two functions that can be used for the composition are and. This is not a unique solution; and also work.

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31 Example 7 – Decomposing Functions Symbolically Each given function is a composite function of the form f (g(x)). Decompose the functions into two functions f (x) and g(x). c.

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32 Example 7 – Solution c. The function f (g(x)) raises 2 to the power 6x – 1. Two functions that can be used for the composition are f (x) = 2 x and g(x) = 6x – 1. Additional solutions exist. cont’d

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33 Example 7 – Decomposing Functions Symbolically Each given function is a composite function of the form f (g(x)). Decompose the functions into two functions f (x) and g(x). d.

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34 Example 7 – Solution d. The function f (g(x)) raises x – 2 to the fourth, third, and second power. Two functions that can be used for the composition are f (x) = x 4 – x 3 + x 2 and g(x) = x – 2. Additional solutions exist. cont’d

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