1-2. Composition of Functions. OBJECTIVES Perform operations with functions. Find composite functions. Iterate functions using real numbers.

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1 - OBJECTIVES Perorm operations with unctions. Find composite unctions. Iterate unctions using real numbers. Composition o Functions BUSINESS Each year, thousands o people visit Yellowstone National Park in Wyoming. Audiotapes or visitors include interviews with early settlers and inormation about the geology, wildlie, and activities o the park. The revenue r () rom the sale o tapes is r() 9.5. Suppose that the unction or the cost o manuacturing tapes is c() What unction could be used to ind the proit on tapes? This problem will be solved in Eample. Real World A p plic atio n To solve the proit problem, you can subtract the cost unction c() rom the revenue unction r(). I you have two unctions, you can orm new unctions by adding, subtracting, multiplying, or dividing the unctions. Z- GRAPHING CALCULATOR EXPLORATION Use a graphing calculator to eplore the sum o two unctions. Enter the unctions () and () 3 as Y and Y, respectively. Enter Y Y as the unction or Y3. To enter Y and Y, press VARS, then select Y-VARS. Then choose the equation name rom the menu. Use TABLE to compare the unction values or Y, Y, and Y3. TRY THESE Use the unctions () and () 3 as Y and Y. Use TABLE to observe the results or each deinition o Y3.. Y3 Y Y. Y3 Y Y 3. Y3 Y Y WHAT DO YOU THINK? 4. Repeat the activity using unctions () and () 5 as Y and Y, respectively. What do you observe? 5. Make conjectures about the unctions that are the sum, dierence, product, and quotient o two unctions. The Graphing Calculator Eploration leads us to the ollowing deinitions o operations with unctions. Operations with Functions Sum: Dierence: Product: Quotient: ( g)() () g() ( g)() () g() ( g)() () g() g ( ) (), g () 0 g ( ) Lesson - Composition o Functions 3

2 For each new unction, the domain consists o those values o common to the domains o and g. The domain o the quotient unction is urther restricted by ecluding any values that make the denominator, g(), zero. Eample Given () 3 4 and g() 4 5, ind each unction. a. ( g)() b. ( g)() ( g)() () g() ( g)() () g() (4 5) c. ( g)() d. g () ( g)() () g() g ( ) (3 4)(4 5) () g ( ) , You can use the dierence o two unctions to solve the application problem presented at the beginning o the lesson. Eample Real World A p plic atio n BUSINESS Reer to the application at the beginning o the lesson. a. Write the proit unction. b. Find the proit on 500, 000, and 5000 tapes. a. Proit is revenue minus cost. Thus, the proit unction p() is p() r() c(). The revenue unction is r() 9.5. The cost unction is c() p() r() c() 9.5 ( ) b. To ind the proit on 500, 000, and 5000 tapes, evaluate p(500), p(000), and p(5000). p(500) 8.7(500) 940 or 40 p(000) 8.7(000) 940 or 6760 p(5000) 8.7(5000) 940 or 4,560 The proit on 500, 000, and 5000 tapes is $40, $6760, and $4,560, respectively. Check by inding the revenue and the cost or each number o tapes and subtracting to ind proit. Functions can also be combined by using composition. In a composition, a unction is perormed, and then a second unction is perormed on the result o the irst unction. You can think o composition in terms o manuacturing a product. For eample, iber is irst made into cloth. Then the cloth is made into a garment. 4 Chapter Linear Relations and Functions

3 In composition, a unction g maps the elements in set R to those in set S. Another unction maps the elements in set S to those in set T. Thus, the range o unction g is the same as the domain o unction. A diagram is shown below. R S S T g() 4 () domain o g() The range o g() is the domain o (). range o () The unction ormed by composing two unctions and g is called the composite o and g. It is denoted by g, which is read as composition g or og. R S T g g() (g()) g [ (g()) g]() Composition o Functions Given unctions and g, the composite unction g can be described by the ollowing equation. [ g]() (g( )) The domain o g includes all o the elements in the domain o g or which g() is in the domain o. Eample 3 Find [ g]() and [g ]() or () 3 8 and g() 5 6. [ g]() (g()) (5 6) Substitute 5 6 or g(). (5 6) 3(5 6) 8 Substitute 5 6 or in (). ( ) [g ]() g(()) g( 3 8) Substitute 3 8 or (). 5( 3 8) 6 Substitute 3 8 or in g() The domain o a composed unction [ g]() is determined by the domains o both () and g(). Lesson - Composition o Functions 5

4 Eample 4 State the domain o [ g]() or () 4 and g(). () 4 Domain: 4 g() Domain: 0 I g() is undeined or a given value o, then that value is ecluded rom the domain o [ g](). Thus, 0 is ecluded rom the domain o [ g](). The domain o () is 4. So or to be in the domain o [ g](), it must be true that g() 4. g() 4 4 g() 4 Multiply each side by. 4 Divide each side by 4. Take the square root o each side. Rewrite the inequality. Thereore, the domain o [ g]() is, 0. The composition o a unction and itsel is called iteration. Each output o an iterated unction is called an iterate. To iterate a unction (), ind the unction value ( 0 ), o the initial value 0. The value ( 0 ) is the irst iterate,. The second iterate is the value o the unction perormed on the output; that is, (( 0 )) or ( ). Each iterate is represented by n, where n is the iterate number. For eample, the third iterate is 3. Eample 5 Find the irst three iterates,,, and 3, o the unction () 3 or an initial value o 0. To obtain the irst iterate, ind the value o the unction or 0. ( 0 ) () () 3 or To obtain the second iterate,, substitute the unction value or the irst iterate,, or. ( ) ( ) ( ) 3 or 5 Now ind the third iterate, 3, by substituting or. 3 ( ) ( 5) ( 5) 3 or 3 Thus, the irst three iterates o the unction () 3 or an initial value o 0 are, 5, and 3. 6 Chapter Linear Relations and Functions

5 C HECK FOR U NDERSTANDING Communicating Mathematics Read and study the lesson to answer each question.. Write two unctions () and g() or which ( g)() 6. Tell how you determined () and g().. Eplain how iteration is related to composition o unctions. 3. Determine whether [ g]() is always equal to [g ]() or two unctions () and g(). Eplain your answer and include eamples or countereamples. 4. Math Journal Write an eplanation o unction composition. Include an everyday eample o two composed unctions and an eample o a realworld problem that you would solve using composed unctions. Guided Practice 5. Given () and g() 9, ind () g(), () g(), () g(), and g (). Practice A Find [ g]() and [g ]() or each () and g(). 6. () 5 7. () 3 g() 3 g() 8. State the domain o [ g]() or () ( ) and g() Find the irst three iterates o the unction () using the initial value Measurement In 954, the Tenth General Conerence on Weights and Measures adopted the kelvin K as the basic unit or measuring temperature or all international weights and measures. While the kelvin is the standard unit, degrees Fahrenheit and degrees Celsius are still in common use in the United States. The unction C(F ) 5 (F 3) relates Celsius temperatures 9 and Fahrenheit temperatures. The unction K(C ) C 73.5 relates Celsius temperatures and Kelvin temperatures. a. Use composition o unctions to write a unction to relate degrees Fahrenheit and kelvins. b. Write the temperatures 40 F, F, 0 F, 3 F, and F in kelvins. E XERCISES Find () g(), () g(), () g(), and g () or each () and g().. () 3. () 3. () 7 g() 9 g() g() 5 4. I () 3 and g(), ind () g(), () g(), () g(), 5 and g (). Lesson - Composition o Functions 7

6 Find [ g]() and [g ]() or each () and g(). B 5. () 9 6. () 7 g() 4 g() 6 7. () 4 8. () g() 3 g() 5 9. () 0. () g() 3 g() 5 6. What are [ g]() and [g ]() or () and g()? C State the domain o [ g]() or each () and g().. () 5 3. () 4. () g() 3 g() 7 g() 4 Find the irst three iterates o each unction using the given initial value. 5. () 9 ; 0 6. () ; 0 7. () (3 ); 0 Applications and Problem Solving Real World A p plic atio n 8. Retail Sara Sung is shopping and inds several items that are on sale at 5% o the original price. The items that she wishes to buy are a sweater originally at $43.98, a pair o jeans or $38.59, and a blouse or $3.99. She has $00 that her grandmother gave her or her birthday. I the sales ta in San Mateo, Caliornia, where she lives is 8.5%, does Sara have enough money or all three items? Eplain. 9. Critical Thinking Suppose the graphs o unctions () and g() are lines. Must it be true that the graph o [ g]() is a line? Justiy your answer. 30. Physics When a heavy bo is being pushed on the loor, there are two dierent orces acting on the movement o the bo. There is the orce o the person pushing the bo and the orce o riction. I W is work in joules, F is orce in newtons, and d is displacement o the bo in meters, W p F p d describes the work o the person, and W F d describes the work created by riction. The increase in kinetic energy necessary to move the bo is the dierence between the work done by the person W p and the work done by riction W. a. Write a unction in simplest orm or net work. b. Determine the net work epended when a person pushes a bo 50 meters with a orce o 95 newtons and riction eerts a orce o 55 newtons. 3. Finance A sales representative or a cosmetics supplier is paid an annual salary plus a bonus o 3% o her sales over $75,000. Let () 75,000 and h() a. I is greater than $75,000, is her bonus represented by [h()] or by h[()]? Eplain. b. Find her bonus i her sales or the year are $400, Critical Thinking Find i [ g]() 4 and g(). 8 Chapter Linear Relations and Functions

7 33. International Business Value-added ta, VAT, is a ta charged on goods and services in European countries. Many European countries oer reunds o some VAT to non-resident based businesses. VAT is included in a price that is quoted. That is, i an item is marked as costing $0, that price includes the VAT. a. Suppose an American company has operations in The Netherlands, where the VAT is 7.5%. Write a unction or the VAT amount paid v(p) i p represents the price including the VAT. b. In The Netherlands, oreign businesses are entitled to a reund o 84% o the VAT on automobile rentals. Write a unction or the reund an American company could epect r(v) i v represents the VAT amount. c. Write a unction or the reund epected on an automobile rental r (p) i the price including VAT is p. d. Find the reunds due on automobile rental prices o $43.8, $5.64, and $ Mied Review 34. Finance The ormula or the simple interest earned on an investment is I prt, where I is the interest earned, p is the principal, r is the interest rate, and t is the time in years. Assume that $5000 is invested at an annual interest rate o 8% and that interest is added to the principal at the end o each year. (Lesson -) a. Find the amount o interest that will be earned each year or ive years. b. State the domain and range o the relation. c. Is this relation a unction? Why or why not? 35. State the relation in the table as a set o ordered pairs. Then state the domain and range o the relation. (Lesson -) 36. What are the domain and the range o the relation {(, 5), (, 6), (3, 7), (4, 8)}? Is the relation a unction? Eplain. (Lesson -) 37. Find g( 4) i g() 3 5. (Lesson -) Given that is an integer, state the relation representing y 3 and 3 by making a table o values. Then graph the ordered pairs o the relation. (Lesson -) 39. SAT/ACT Practice Find (n ) i () 9. A n n 9 B n n 8 C n 5n D 9 E n 4n 8 y Etra Practice See p. A6. Lesson - Composition o Functions 9