Inverse Functions; Eponential and Logarithmic Functions Outline 4-1 Operations on Functions; Composition 4-2 Inverse Functions 4-3 Eponential Functions 4-4 The Eponential Function with Base e 4- Logarithmic Functions 4-6 Common and Natural Logarithms 4-7 Eponential and Logarithmic Equations Chapter 4 Group Activit: Comparing Regression Models Chapter 4 Review Cumulative Review Eercise for Chapters 3 and 4 Application You have just inherited a trust fund of $18,000 that ou cannot access for 8 more ears. You are hoping that this investment will double b then, as ou have plans to purchase a sports car for $3,000. The Rule of 72 can be used to quickl determine how long it will take an investment to double at a given interest rate. If r is the annual rate, then the Rule of 72 states that an investment at this rate will double in approimatel 72/(100r) ears. If our trust fund is invested at 10.4% compounded annuall, how accurate is the Rule of 72 approimation? 40,000 3,000 30,000 2,000 20,000 1,000 10,000,000 0
Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational functions. The general class of functions defined b means of the algebraic operations of addition, subtraction, multiplication, division, and the taking of powers and roots on variables and constants are called algebraic functions. In the first two sections of this chapter, we discuss some operations that can be performed on functions to produce new functions, including the ver important concept of an inverse function. Net, we define and investigate the properties of two new and important tpes of functions called eponential functions and logarithmic functions. These functions are not algebraic, but are members of another class of functions called transcendental functions. The eponential functions and logarithmic functions are used in describing and solving a wide variet of real-world problems, including growth of populations of people, animals, and bacteria; radioactive deca; growth of mone at compound interest; absorption of light as it passes through air, water, or glass; and magnitudes of sounds and earthquakes. We consider applications in these areas plus man more in the sections that follow. Preparing for This Chapter Before getting started on this chapter, review the following concepts: Eponents (Appendi A, Sections and 6) Functions (Chapter 1, Section 3) Graphs of Functions (Chapter 1, Section 4) Quadratic Equations (Chapter 2, Section ) Equation Solving Techniques (Chapter 2, Section 6) Section 4-1 Operations on Functions; Composition Operations on Functions Composition Applications 244 If two functions f and g are both defined at a real number, and if f() and g() are both real numbers, then it is possible to perform real number operations such as addition, subtraction, multiplication, or division with f() and g(). Furthermore, if g() is a number in the domain of f, then it is also possible to evaluate f at g(). In this section we see how operations on the values of functions can be used to define operations on the functions themselves.
4-1 Operations on Functions; Composition 24 Operations on Functions The functions f and g given b f() 2 3 and g() 2 4 are defined for all real numbers. Thus, for an real we can perform the following operations: f() g() 2 3 2 4 2 2 1 f() g() 2 3 ( 2 4) 2 2 7 f()g() (2 3)( 2 4) 2 3 3 2 8 12 For 2 we can also form the quotient f() 2 3 g() 2 4 2 Notice that the result of each operation is a new function. Thus, we have ( f g)() f() g() 2 2 1 ( f g)() f() g() 2 2 7 ( fg)() f()g() 2 3 3 2 8 12 g f f() 2 3 () g() 2 4 2 Sum Difference Product Quotient Notice that the sum, difference, and product functions are defined for all values of, as were f and g, but the domain of the quotient function must be restricted to eclude those values where g() 0. DEFINITION 1 OPERATIONS ON FUNCTIONS The sum, difference, product, and quotient of the functions f and g are the functions defined b ( f g)() f() g() ( f g)() f() g() ( fg)() f()g() g f f() () g() g() 0 Sum function Difference function Product function Quotient function Each function is defined on the intersection of the domains of f and g, with the eception that the values of where g() 0 must be ecluded from the domain of the quotient function.
246 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Eplore/Discuss 1 FIGURE 1 Graphing sum, difference, product, and quotient functions. Enter 1 4 and 2 3 in the equation editor of a graphing utilit [Fig. 1(a)], graph 1 and 2 in the same viewing window* [Fig. 1(b)], and use Trace to determine the values of for which each function is defined. Use Trace in Figures 1(c) through 1(f) to determine the domains of the corresponding functions. 4.7 4.7 (a) Equation editor (b) 1 and 2 4.7 4.7 4.7 4.7 (c) 3 (d) 4 4.7 4.7 4.7 4.7 (e) (f) 6 EXAMPLE 1 Solution Finding the Sum, Difference, Product, and Quotient Functions Let f() 4 and g() 3. Find the functions f g, f g, fg, and f/g, and find their domains. ( f g)() f() g() 4 3 ( f g)() f() g() 4 3 ( fg)() f()g() 4 3 (4 )(3 ) 12 2 g f f() 4 () g() 3 4 3 *It is convenient to choose Xmin and Xma so that the piels have one-decimal-place screen coordinates. See Problems 3 and 36 in Eercise 1-2 or consult our manual.
4-1 Operations on Functions; Composition 247 3 0 4 [ Domain of f Domain of g 3 0 4 Domain of f g, f g, and fg [ 3 0 4 f Domain of g [ [ ) 3 0 4 [ The domains of f and g are Domain of f: 4 or (, 4] Domain of g: 3 or [ 3, ) The intersection of these domains is (, 4] [ 3, ) [ 3, 4] This is the domain of the functions f g, f g, and fg. Since g( 3) 0, 3 must be ecluded from the domain of the quotient function. Thus, Domain of f : ( 3, 4] g MATCHED PROBLEM 1 Let f() and g() 10. Find the functions f g, f g, fg, and f/g, and find their domains. Composition Consider the function h given b the equation Inside the radical is a first-degree polnomial that defines a linear function. So the function h is reall a combination of a square root function and a linear function. We can see this more clearl as follows. Let Then h() 2 1 u 2 1 g() u f(u) h() f(g()) The function h is said to be the composite of the two functions f and g. (Loosel speaking, we can think of h as a function of a function.) What can we sa about the domain of h given the domains of f and g? In forming the composite h() f(g()): must be restricted so that is in the domain of g and g() is in the domain of f. Since the domain of f, where f(u) u, is the set of nonnegative real numbers, we see that g() must be nonnegative; that is, g() 0 2 1 0 1 2 Thus, the domain of h is this restricted domain of g.
248 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS A special function smbol is often used to represent the composite of two functions, which we define in general terms below. DEFINITION 2 COMPOSITE FUNCTIONS Given functions f and g, then f g is called their composite and is defined b the equation ( f g)() f(g()) The domain of f g is the set of all real numbers in the domain of g where g() is in the domain of f. As an immediate consequence of Definition 2, we have (see Fig. 2): The domain of f g is alwas a subset of the domain of g, and the range of f g is alwas a subset of the range of f. FIGURE 2 Composite functions. Domain f g f g (f g)() f [g()] Range f g g f g() Domain g Range g Domain f Range f EXAMPLE 2 Finding the Composition of Two Functions Find ( f g)() and (g f )() and their domains for f() 10 and g() 3 4 1. Solution ( f g)() f(g()) f(3 4 1) (3 4 1) 10 (g f )() g( f()) g( 10 ) 3( 10 ) 4 1 3 40 1 The functions f and g are both defined for all real numbers. If is an real number, then is in the domain of g, g() is in the domain of f, and, consequentl, is in the domain of f g. Thus, the domain of f g is the set of all real numbers. Using similar reasoning, the domain of g f also is the set of all real numbers. MATCHED PROBLEM 2 Find ( f g)() and (g f )() and their domains for f() 2 1 and g() ( 1)/2. If two functions are both defined for all real numbers, then so is their composition.
4-1 Operations on Functions; Composition 249 Eplore/Discuss 2 Verif that if f() 1/(1 2) and g() 1/, then ( f g)() /( 2). Clearl, f g is not defined at 2. Are there an other values of where f g is not defined? Eplain. If either function in a composition is not defined for some real numbers, then, as Eample 3 illustrates, the domain of the composition ma not be what ou first think it should be. EXAMPLE 3 Solution Finding the Composition of Two Functions Find ( f g)() and its domain for f() 4 2 and g() 3. We begin b stating the domains of f and g, a good practice in an composition problem: Domain f: 2 2 or [ 2, 2] Domain g: 3 or (, 3] Net we find the composition: ( f g)() f(g()) f( 3 4 ( 3 ) 2 4 (3 ) 1 ( t) 2 t, t 0 Even though 1 is defined for all 1, we must restrict the domain of f g to those values that also are in the domain of g. Thus, Domain f g: 1 and 3 or [ 1, 3] MATCHED PROBLEM 3 Find ( f g)() and its domain for f() 9 2 and g() 1. CAUTION The domain of f g cannot alwas be determined simpl b eamining the final form of ( f g)(). An numbers that are ecluded from the domain of g must also be ecluded from the domain of f g.
20 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Eplore/Discuss 3 Refer to Eample 3. Enter 1 4 2, 2 3, and 3 1 ( 2 ()) in the equation editor of our graphing utilit and graph 3. Does this graph agree with the answer we found in Eample 3? Does our graphing utilit seem to handle composition correctl? (Not all do!) Applications In calculus, it is not onl important to be able to find the composition of two functions, but also to recognize when a given function is the composition of two simpler functions. EXAMPLE 4 Recognizing Composition Forms Epress h as a composition of two simpler functions for h() (3 ) Solution If we let f() and g() 3, then h() (3 ) f(3 ) f(g()) ( f g)() and we have epressed h as the composition of f and g. MATCHED PROBLEM 4 Epress h as a composition of the square root function and a linear function for h() 4 7. You will encounter the operations discussed in this section in man different situations. The net eample shows how these operations are used in economic analsis. EXAMPLE Market Research The research department for an electronics firm estimates that the weekl demand for a certain brand of audiocassette plaers is given b f( p) 20,000 1,000p Demand function where is the number of cassette plaers retailers are likel to bu per week at $p per plaer. The research department also has determined that the total cost (in dollars) of producing cassette plaers per week is given b C() 7,000 4 Cost function
4-1 Operations on Functions; Composition 21 and the total weekl revenue (in dollars) obtained from the sale of these cassette plaers is given b R() 20 1 1,000 2 Revenue function Epress the firm s weekl profit as a function of the price p. Solution Since profit is revenue minus cost, the profit function is the difference of the revenue and cost functions, P R C. Since R and C are given as functions of, we first epress P as a function of : P() (R C)() R() C() 20 1 1,000 2 (7,000 4) 16 1 1,000 2 7,000 Net, we use composition to epress P as a function of the price p: (P f )( p) P( f( p)) P(20,000 1,000p) 16(20,000 1,000p) 1 1,000 (20,000 1,000p)2 7,000 320,000 16,000p 400,000 40,000p 1,000p 2 7,000 1,000 24,000p 1,000p 2 Technicall, P f and P are different functions since the first has independent variable p and the second has independent variable. However, since both functions represent the same quantit, it is customar to use the same smbol to name each function. Thus, P( p) 1,000 24,000p 1,000p 2 epresses the weekl profit P as a function of price p. MATCHED PROBLEM Repeat Eample for the functions f(p) 10,000 1,000p C() 90,000 R() 10 1 1,000 2
22 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answers to Matched Problems 1. ( f g)() 10 ; ( f g)() 10 ; ( fg)() 10 2 ; ( f/g)() /(10 ) ; the functions f g, f g, and fg have domain [0, 10], the domain of f/g is [0, 10) 2. ( f g)(), domain (, ); (g f )(), domain (, ) 3. ( f g)() 10 ; domain: 1 and 10 or [1, 10] 4. h() ( f g)(), where f() and g() 4 7. P( p) 140,000 1,000p 1,000p 2 EXERCISE 4-1 A Problems 1 10 refer to the graphs of f and g shown below. f() g() 18. f() 2 ; g() 3 2 4 19. f() 1 ; g() 2 3 4 20. f() ; g() 3 2 21. f() 1/3 ; g() 2 3 4 22. f() 2/3 ; g() 8 3 B In Problems 23 26, find f g and g f. Graph f, g, f g, and g f in a squared viewing window and describe an apparent smmetr between these graphs. 1. Construct a table of values of ( f g)() for 3, 2, 1, 0, 1, 2, and 3, and sketch the graph of f g. 2. Construct a table of values of (g f )() for 3, 2, 1, 0, 1, 2, and 3, and sketch the graph of g f. 23. 24. 2. 26. f() 1 2 1; g() 2 2 f() 3 2; g() 1 3 2 3 f() 2 3 3 ; g() 3 2 2 f() 2 3; g() 1 2 3 2 Use the graphs of f and g to find each of the following: 3. ( f g)( 1) 4. ( f g)(2). (g f )( 2) 6. (g f )(3) 7. f(g(1)) 8. f(g(0)) 9. g( f(2)) 10. g( f( 3)) In Problems 11 16, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains. 11. f() 4; g() 1 12. f() 3; g() 2 13. f() 2 2 ; g() 2 1 14. f() 3; g() 2 4 1. f() 3 ; g() 2 1 16. f() 2 7; g() 9 2 In Problems 17 22, for the indicated functions f and g, find the functions f g and g f, and find their domains. 17. f() 3 ; g() 2 1 In Problems 27 32, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains. 27. 28. 29. 30. 31. 32. In Problems 33 38, for the indicated functions f and g, find the functions f g and g f, and find their domains. 33. 34. 3. f() 2; g() 1 36. f() 3; g() 1 2 37. f() 2 ; g() 3 f() 4; g() 3 f() 2; g() 4 f() 1 ; g() 2 f() 2 6; g() 7 6 2 f() 8 2 2 ; g() 2 7 10 f() ; g() 4 f() ; g() 2 f() 1 ; g() 1 38. f() 1 ; g() 1
4-1 Operations on Functions; Composition 23 Use the graphs of functions f and g shown below to match each function in Problems 39 44 with one of graphs (a) (f). 39. ( f g)() 40. ( f g)() 41. (g f )() 42. ( fg)() 43. 44. f() f g () g() g f () 4. h() (2 7) 4 46. h() (3 ) 7 47. h() 4 2 48. h() 3 11 49. h() 3 7 0. h() 6 3 1. h() 4 2. h() 2 3 1 3. Are the functions fg and gfidentical? Justif our answer. 4. Are the functions f g and g f identical? Justif our answer.. Is there a function g that satisfies f g g f f for all functions f? If so, what is it? 6. Is there a function g that satisfies fg gf f for all functions f? If so, what is it? In Problems 7 60, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains. 7. f() 1 ; g() 1 8. f() 1; g() 6 1 (a) (b) 9. f() 1 ; g() 1 60. f() ; g() In Problems 61 66, for the indicated functions f and g, find the functions f g and g f, and find their domains. 61. f() 4 ; g() 2 62. 63. f() 1; g() 2 f() ; g() 2 (c) (d) 64. f() 2 4 ; g() 1 6. f() 2 2 ; g() 9 2 66. f() 2 9; g() 2 2 (e) In Problems 4 2, epress h as a composition of two simpler functions f and g of the form f() n and g() a b, where n is a rational number and a and b are integers. (f) In Problems 67 72, enter the given epression for (f g)() eactl as it is written and graph on a graphing utilit for 10 10. Then simplif the epression, enter the result, and graph in a new viewing window, again for 10 10. Find the domain of f g. Which is the correct graph of f g? 67. 68. f() 2 ; g() 3 ; ( f g)() ( 3 ) 2 f() 6 2 ; g() 1; ( f g)() 6 ( 1) 2
24 4 INVERSE FUNCTIONS; EXPONENTIAL AND LOGARITHMIC FUNCTIONS 69. 70. 71. 72. f() 2 ; g() 2 4; ( f g)() ( 2 4) 2 f() 2 ; g() 4 2 ; ( f g) ( 4 2 ) 2 f() 2 7; g() 9 2 ; ( f g)() ( 9 2 ) 2 7 f() 2 7; g() 2 9; ( f g)() ( 2 9) 2 7 APPLICATIONS 73. Market Research. The demand and the price p (in dollars) for a certain product are related b f(p) 4,000 200p The revenue (in dollars) from the sale of units is given b R() 20 1 200 2 and the cost (in dollars) of producing units is given b C() 10 30,000 Epress the profit as a function of the price p. 74. Market Research. The demand and the price p (in dollars) for a certain product are related b f(p),000 100p The revenue (in dollars) from the sale of units and the cost (in dollars) of producing units are given, respectivel, b R() 0 1 and C() 20 40,000 100 2 Epress the profit as a function of the price p. 7. Pollution. An oil tanker aground on a reef is leaking oil that forms a circular oil slick about 0.1 foot thick (see the figure). The radius of the slick (in feet) t minutes after the leak first occurred is given b r(t) 0.4t 1/3 Epress the volume of the oil slick as a function of t. (A) Epress the distance d between the balloon and the observer as a function of the balloon s distance h above the ground. (B) If the balloon s distance above ground after t seconds is given b h t, epress the distance d between the balloon and the observer as a function of t. 77. Fluid Flow. A conical paper cup with diameter 4 inches and height 4 inches is initiall full of water. A small hole is made in the bottom of the cup and the water begins to flow out of the cup. Let h and r be the height and radius, respectivel, of the water in the cup t minutes after the water begins to flow. (A) Epress r as a function of h. (B) Epress the volume V as a function of h. (C) If the height of the water after t minutes is given b h(t) 0. t epress V as a function of t. 78. Evaporation. A water trough with triangular ends is 6 feet long, 4 feet wide, and 2 feet deep. Initiall, the trough is full of water, but due to evaporation, the volume of the water is decreasing. Let h and w be the height and width, respectivel, of the water in the tank t hours after it began to evaporate. 6 feet 4 inches h r 1 V r 2 h 3 4 inches 4 feet r 2 feet h w V 3wh A r 2 V 0.1A 76. Weather Balloon. A weather balloon is rising verticall. An observer is standing on the ground 100 meters from the point where the weather balloon was released. (A) Epress w as a function of h. (B) Epress V as a function of h. (C) If the height of the water after t hours is given b h(t) 2 0.2 t epress V as a function of t.