5.1 Composite Functions

Transcription

1 SECTION. Composite Funtions 7. Composite Funtions PREPARING FOR THIS SECTION Before getting started, review the following: Find the Value of a Funtion (Setion., pp. 9 ) Domain of a Funtion (Setion., pp. ) Now Work the Are You Prepared? problems on page. OBJECTIVES Form a Composite Funtion (p. 7) Find the Domain of a Composite Funtion (p. 8) Figure Form a Composite Funtion Suppose that an oil tanker is leaking oil and you want to determine the area of the irular oil path around the ship. See Figure. It is determined that the oil is leaking from the tanker in suh a way that the radius of the irular path of oil around the ship is inreasing at a rate of feet per minute. Therefore, the radius r of the oil path at any time t, in minutes, is given by rt = t. So after 0 minutes the radius of the oil path is r0 = 0 = 60 feet. The area A of a irle as a funtion of the radius r is given by Ar = pr. The area of the irular path of oil after 0 minutes is A60 = p60 = 600p square feet. Notie that 60 = r0, so A60 = Ar0. The argument of the funtion A is the output of a funtion! In general, we an find the area of the oil path as a funtion of time t by evaluating Art and obtaining Art = At = pt = 9pt. The funtion Art is a speial type of funtion alled a omposite funtion. As another eample, onsider the funtion y = +. If we write y = fu = u and u = g = +, then, by a substitution proess, we an obtain the original funtion: y = fu = fg = +. In general, suppose that f and g are two funtions and that is a number in the domain of g. By evaluating g at, we get g. If g is in the domain of f, then we may evaluate f at g and obtain the epression fg. The orrespondene from to fg is alled a omposite funtion f g. DEFINITION Given two funtions f and g, the omposite funtion, denoted by f g (read as f omposed with g ), is defined by f g = fg The domain of f g is the set of all numbers in the domain of g suh that g is in the domain of f. Look arefully at Figure. Only those s in the domain of g for whih g is in the domain of f an be in the domain of f g. The reason is that if g is not in the domain of f then fg is not defined. Beause of this, the domain of f g is a subset of the domain of g; the range of f g is a subset of the range of f. Figure Domain of g g Range of g g () Domain of f Range of f g g () f f(g ()) Domain of f g Range of f g f g

2 8 CHAPTER Eponential and Logarithmi Funtions Figure provides a seond illustration of the definition. Here is the input to the funtion g, yielding g. Then g is the input to the funtion f, yielding fg. Notie that the inside funtion g in fg is done first. Figure INPUT g g () f OUTPUT f(g ()) Figure EXAMPLE Evaluating a Composite Funtion Suppose that f = - and g =. Find: (a) f g (b) g f () f f- (d) g g- (a) (b) () (d) f g = fg = f = # - = 9 g() = f() = - g() = g f = gf = g- = # - = - f() = - g() = f() = - f f- = ff- = f = # - = 7 f(- ) = (- ) - = g g- = gg- = g- = # - = - 6 g(- ) = - COMMENT Graphing alulators an be used to evaluate omposite funtions.* Let Y = f() = - and Y = g() =. Then, using a TI-8 Plus graphing alulator, (f g)() isfoundasshowninfigure. Notie that this is the result obtained in Eample (a). Now Work PROBLEM Find the Domain of a Composite Funtion EXAMPLE Finding a Composite Funtion and Its Domain Suppose that f = + - and g = +. Find: (a) f g (b) g f Then find the domain of eah omposite funtion. The domain of f and the domain of g are the set of all real numbers. (a) f g = fg = f + = ) - f() = + - = = Sine the domains of both f and g are the set of all real numbers, the domain of f g is the set of all real numbers. *Consult your owner s manual for the appropriate keystrokes.

3 SECTION. Composite Funtions 9 (b) g f = gf = g + - = g() = + = = Sine the domains of both f and g are the set of all real numbers, the domain of g f is the set of all real numbers. Look bak at Figure on page 7. In determining the domain of the omposite funtion f g = fg, keep the following two thoughts in mind about the input.. Any not in the domain of g must be eluded.. Any for whih g is not in the domain of f must be eluded. EXAMPLE EXAMPLE Finding the Domain of f g Find the domain of f g if f = and g =. + - For f g = fg, first note that the domain of g is ƒ Z 6, so elude from the domain of f g. Net note that the domain of f is ƒ Z - 6, whih means that g annot equal -. Solve the equation g = - to determine what additional value(s) of to elude. - = - Also elude - from the domain of f g. The domain of f g is ƒ Z -, Z 6. Chek: For =, g = is not defined, so f g = fg is not defined. - For =-, g- = =-, and f g- = fg- = f- - is not defined. Now Work PROBLEM = - - = - + = - = - g() = - Finding a Composite Funtion and Its Domain Suppose that f = and g =. + - Find: (a) f g (b) f f Then find the domain of eah omposite funtion. The domain of f is ƒ Z - 6 and the domain of g is ƒ Z 6. (a) f g = fg = fa = - b f() = = = - + = - + Multiply by - -. In Eample, we found the domain of f g to be ƒ Z -, Z 6.

4 0 CHAPTER Eponential and Logarithmi Funtions We ould also find the domain of f g by first looking at the domain of g: ƒ Z 6. We elude from the domain of f g as a result. Then we look at f g and notie that annot equal -, sine = - results in division by 0. So we also elude - from the domain of f g. Therefore, the domain of f g is ƒ Z -, Z 6. + (b) f f = ff = fa = + + = + + b = f() = + The domain of f f onsists of those in the domain of f, ƒ Z - 6, for whih f = + Z - + = - = - ( + ) = - - = - = - Multiply by + +. or, equivalently, Z - The domain of f f is e ` Z -, Z - f. We ould also find the domain of f f by reognizing that - is not in the domain of f and so should be eluded from the domain of f f. Then, looking at f f, we see that annot equal - Do you see why? Therefore, the. domain of f f is e ` Z -, Z - f. Now Work PROBLEMS AND Look bak at Eample, whih illustrates that, in general, Sometimes f g does equal g f, as shown in the net eample. f g Z g f. EXAMPLE Showing That Two Composite Funtions Are Equal If f = - and g = +, show that f g = g f = for every in the domain of f g and g f. f g = fg = fa + b = a + b - = + - = g() = + ( + ) = Substitute g() into the rule for f, f() = -.

5 SECTION. Composite Funtions Seeing the Conept Using a graphing alulator, let Y = f() = - Y = g() = ( + ) Y = f g, Y = g f g f = gf = g - = - + = = f() = - Substitute f() into the rule for g, g() = ( + ). Using the viewing window -, - y, graph only Y and Y. What do you see? TRACE to verify that Y = Y. We onlude that f g = g f =. In Setion., we shall see that there is an important relationship between funtions f and g for whih f g = g f =. Now Work PROBLEM Calulus Appliation Some tehniques in alulus require that we be able to determine the omponents of a omposite funtion. For eample, the funtion H = + is the omposition of the funtions f and g, where f = and g = +, beause H = f g = fg = f + = +. EXAMPLE 6 Finding the Components of a Composite Funtion Find funtions f and g suh that f g = H if H = + 0. The funtion H takes + and raises it to the power 0. A natural way to deompose H is to raise the funtion g = + to the power 0. If we let f = 0 and g = +, then Figure g f f g = fg f(g ()) = f( + ) = f + g () = = ( + ) 0 + H() = ( + ) 0 = + 0 = H H See Figure. Other funtions f and g may be found for whih f g = H in Eample 6. For eample, if f = and g = +, then f g = fg = f + = + = + 0 Although the funtions f and g found as a solution to Eample 6 are not unique, there is usually a natural seletion for f and g that omes to mind first. EXAMPLE 7 Finding the Components of a Composite Funtion Find funtions f and g suh that f g = H if H = Here H is the reiproal of g = +. If we let f = and g = +, we find that f g = fg = f + = + +. = H Now Work PROBLEM

6 CHAPTER Eponential and Logarithmi Funtions. Assess Your Understanding Are You Prepared? Answers are given at the end of these eerises. If you get a wrong answer, read the pages listed in red.. Find f if f = - +. (pp. 9 ). Find the domain of the funtion. Find f if f = -. (pp. 9 ) (pp. ) f = - -. Conepts and Voabulary. Given two funtions f and g, the, denoted f g, is defined by f g() =.. True or False fg = f # g(). 6. True or False The domain of the omposite funtion f g is the same as the domain of g. Skill Building In Problems 7 and 8, evaluate eah epression using the values given in the table (a) f g (b) f g- f () () g f- (d) g f0 g() (e) g g- (f) f f (a) f g (b) f g f () () g f (d) g f g() (e) g g (f) f f In Problems 9 and 0, evaluate eah epression using the graphs of y = f and y = g shown in the figure. 9. (a) g f- (b) g f0 y () f g- (d) f g y g () 6 (6, ) (7, ) 0. (a) g f (b) g f (, ) (, ) (8, ) () f g0 (d) f g (, ) (, ) (7, ) (, ) (, ) (, ) (6, ) (, ) 6 8 y f () (, ) (, ) In Problems 0, for the given funtions f and g, find: (a) f g (b) g f () f f (d) g g0. f = ; g = +. f = + ; g = -. f = - ; g = -. f = ; g = -. f = ; g = 6. f = + ; g = 7. f = ƒƒ; g = + 9. f = 8. f = ƒ - ƒ; g = + + ; g = 0. f = > ; g = + In Problems 8, find the domain of the omposite funtion f g.. f = - ; g =. f = + ; g =-

7 SECTION. Composite Funtions. f = - ; g =-. f = + ; g =. f = ; g = + 6. f = - ; g = - 7. f = + ; g = - 8. f = + ; g = - In Problems 9, for the given funtions f and g, find: (a) f g (b) g f () f f (d) State the domain of eah omposite funtion. 9. f = + ; g = g g 0. f = - ; g = -. f = + ; g =. f = + ; g = +. f = ; g = +. f = + ; g = +. f = 7. f = - ; g = - ; g =- 6. f = 8. f = + ; g =- + ; g = 9. f = ; g = + 0. f = - ; g = -. f = + ; g = -. f = + ; g = -. f = - + ; g = + - In Problems, show that f g = g f =.. f = - - ; g = + -. f = ; g = 6. f = ; g = 7. f = ; g = 8. f = + ; g = - 9. f = - 6; g = f = - ; g = -. f = a + b; g = a - b a Z 0. f = ; g = In Problems 8, find funtions f and g so that f g = H.. H = +. H = +. H = + 6. H = - 7. H = ƒ + ƒ 8. H = ƒ + ƒ Appliations and Etensions 9. If f = and g =, find f g 6. Surfae Area of a Balloon The surfae area S (in square and g f. meters) of a hot-air balloon is given by 60. If f = + find f f. -, 6. If f = + and g = + a, find a so that the graph of f g rosses the y-ais at. 6. If f = - 7 and g = + a, find a so that the graph of f g rosses the y-ais at 68. In Problems 6 and 6, use the funtions f and g to find: (a) f g (b) g f () the domain of f g and of g f (d) the onditions for whih f g = g f 6. f = a + b; g = + d 6. f = a + b ; g = m + d where r is the radius of the balloon (in meters). If the radius r is inreasing with time t (in seonds) aording to the formula rt = t, t Ú 0, Sr = pr balloon as a funtion of the time t. find the surfae area S of the 66. Volume of a Balloon The volume V (in ubi meters) of the hot-air balloon desribed in Problem 6 is given by V r = If the radius r is the same funtion of t as in pr. Problem 6, find the volume V as a funtion of the time t. 67. Automobile Prodution The number N of ars produed at a ertain fatory in one day after t hours of operation is given by Nt = 00t - t, 0 t 0. If the ost C

8 CHAPTER Eponential and Logarithmi Funtions (in dollars) of produing N ars is CN =, N, 7. Foreign Ehange Traders often buy foreign urreny in find the ost C as a funtion of the time t of operation of the fatory. hope of making money when the urreny s value hanges. For eample, on June, 009, one U.S. dollar ould purhase 68. Environmental Conerns The spread of oil leaking from a 0.7 Euros, and one Euro ould purhase 7.0 yen. tanker is in the shape of a irle. If the radius r (in feet) of the Let f represent the number of Euros you an buy with spread after t hours is rt = 00t, find the area A of the dollars, and let g represent the number of yen you an oil slik as a funtion of the time t. buy with Euros. (a) Find a funtion that relates dollars to Euros. (b) Find a funtion that relates Euros to yen. () Use the results of parts (a) and (b) to find a funtion 69. Prodution Cost The prie p,in dollars,of a ertain produt and the quantity sold obey the demand equation p = Suppose that the ost C, in dollars, of produing units is C = Assuming that all items produed are sold, find the ost C as a funtion of the prie p. [Hint: Solve for in the demand equation and then form the omposite.] 70. Cost of a Commodity The prie p, in dollars,of a ertain ommodity and the quantity sold obey the demand equation p = Suppose that the ost C, in dollars, of produing units is C = Assuming that all items produed are sold, find the ost C as a funtion of the prie p. 7. Volume of a Cylinder The volume V of a right irular ylinder of height h and radius r is V = pr h. If the height is twie the radius, epress the volume V as a funtion of r. 7. Volume of a Cone The volume V of a right irular one is V = If the height is twie the radius, epress the pr h. volume V as a funtion of r. Are You Prepared? Answers ƒ Z -, Z 6 that relates dollars to yen. That is, find g f = gf. (d) What is gf000? 7. Temperature Conversion The funtion C(F) = (F - ) 9 onverts a temperature in degrees Fahrenheit, F,to a temperature in degrees Celsius, C. The funtion KC = C + 7, onverts a temperature in degrees Celsius to a temperature in kelvins, K. (a) Find a funtion that onverts a temperature in degrees Fahrenheit to a temperature in kelvins. (b) Determine 80 degrees Fahrenheit in kelvins. 7. Disounts The manufaturer of a omputer is offering two disounts on last year s model omputer. The first disount is a $00 rebate and the seond disount is 0% off the regular prie, p. (a) Write a funtion f that represents the sale prie if only the rebate applies. (b) Write a funtion g that represents the sale prie if only the 0% disount applies. () Find f g and g f. What does eah of these funtions represent? Whih ombination of disounts represents a better deal for the onsumer? Why? 76. If f and g are odd funtions, show that the omposite funtion f g is also odd. 77. If f is an odd funtion and g is an even funtion, show that the omposite funtions f g and g f are both even.. One-to-One Funtions; Inverse Funtions PREPARING FOR THIS SECTION Before getting started, review the following: Funtions (Setion., pp. 6 ) Inreasing/Dereasing Funtions (Setion., pp. 70 7) Now Work the Are You Prepared? problems on page 6. Rational Epressions (Appendi A, Setion A., pp. A6 A) OBJECTIVES Determine Whether a Funtion Is One-to-One (p. ) Determine the Inverse of a Funtion Defined by a Map or a Set of Ordered Pairs (p. 7) Obtain the Graph of the Inverse Funtion from the Graph of the Funtion (p. 9) Find the Inverse of a Funtion Defined by an Equation (p. 60)