SECTION 25 Combining Functions


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1 2 Combining Functions Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps is 80 feet. The trajector of the ccle through the air is given b the graph of f() v 2 2 where v is the velocit of the ccle in feet per second as it leaves the ramp. 92. Phsics. The trajector of a circus performer shot from a cannon is given b the graph of the function Both the cannon and the net are 10 feet high (see the figure). f() f() f() 10 feet 10 feet 80 feet (A) How fast must the ccle be traveling when it leaves the ramp in order to follow the trajector illustrated in the figure? (B) What is the maimum height of the ccle above the ground as it follows this trajector? (A) How far from the muzzle of the cannon should the center of the net be placed so that the performer lands in the center of the net? (B) What is the maimum height of the performer above the ground? SECTION 2 Combining Functions Operations on Functions Composition Elementar Functions Vertical and Horizontal Shifts Reflections, Epansions, and Contractions 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. We also investigate the graphic implications of some of these operations. 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() f() g() 2 3 ( 2 4) f()g() (2 3)( 2 4)
2 166 2 Graphs and Functions For 2 we can also form the quotient f() 2 3 g() Notice that the result of each operation is a new function. Thus, we have ( f g)() f() g() ( f g)() f() g() ( fg)() f()g() g f f() 2 3 () g() 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. EXAMPLE 1 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. Solution ( f g)() f() g() 4 3 ( f g)() f() g() 4 3 ( fg)() f()g() 4 3 (4 )(3 ) 12 2
3 2 Combining Functions Domain of g [ Domain of f [ The domains of f and g are The intersection of these domains is g f f() 4 () g() Domain of f: 4 or (, 4] Domain of g: 3 or [ 3, ) Domain of f g, f g, and fg [ f Domain of g [ ) [ (, 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 h() 2 1 Inside the radical is a firstdegree 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 u 2 1 g() u f(u) Then 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() Thus, the domain of h is this restricted domain of g.
4 168 2 Graphs and 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. 1): 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 1 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() Solution ( f g)() f [g()] f(3 4 1) (3 4 1) 10 (g f )() g[ f()] g( 10 ) 3( 10 ) 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.
5 2 Combining Functions 169 EXPLOREDISCUSS 1 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 Finding the Composition of Two Functions Find ( f g)() and its domain for f() 4 2 and g() 3. Solution 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. 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.
6 170 2 Graphs and 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. Elementar Functions The functions g() 2 4 h() ( 4) 2 k() 4 2 can all be obtained from the function f() 2 b performing simple operations on f: g() f() 4 h() f( 4) k() 4f() It follows that the graphs of functions g, h, and k are closel related to the graph of function f. Before eploring relationships of this tpe, we want to identif some elementar functions, summarize their basic properties, and include them in our librar of elementar functions. Figure 2 shows si basic functions that ou will encounter frequentl. You should know the definition, domain, and range of each and be able to sketch their graphs. FIGURE 2 Some basic functions and their graphs. [Note: Letters used to designate these functions ma var from contet to contet; R is the set of all real numbers.] f() g() h() Identit function Absolute value function Square function f () g() h() 2 Domain: R Domain: R Domain: R Range: R Range: [0, ) Range: [0, ) (a) (b) (c)
7 2 Combining Functions 171 m() n() p() Cube function Square root function Cube root function m() 3 n() p() 3 Domain: R Domain: [0, ) Domain: R Range: R Range: [0, ) Range: R (d) (e) (f) Vertical and Horizontal Shifts How are the graphs of ( f g)(), ( fg)(), and ( f g)() related to the graphs of f() and g()? In general, this is a difficult question to answer. However, if g is chosen to be a ver simple function, such as g() k or g() h, then we can establish some ver useful relationships between the graph of f() and the graphs of f() k, kf(), and f( h). We refer to the graph obtained b performing one of these operations on a function f as a transformation of the graph of f(). f() EXPLOREDISCUSS 2 Let. (A) Graph f() k for k 2, 0, and 1 simultaneousl in the same coordinate sstem. Describe the relationship between the graph of f() and the graph of f() k for k, an real number. (B) Graph f( h) for h 2, 0, and 1 simultaneousl in the same coordinate sstem. Describe the relationship between the graph of f() and the graph of f( h) for h, an real number. EXAMPLE Vertical and Horizontal Shifts (A) How are the graphs of 2 2 and 2 3 related to the graph of 2? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How are the graphs of ( 2) 2 and ( 3) 2 related to the graph of 2? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. Solutions (A) The graph of 2 2 is the same as the graph of 2 shifted upward 2 units, and the graph of 2 3 is the same as the graph of 2 shifted downward 3 units. Figure 3 on the net page confirms these conclusions. [It appears that the graph of f() k is the graph of f() shifted up if k is positive and down if k is negative.]
8 172 2 Graphs and Functions FIGURE 3 Vertical shifts (B) The graph of ( 2) 2 is the same as the graph of 2 shifted to the left 2 units, and the graph of ( 3) 2 is the same as the graph of 2 shifted to the right 3 units. Figure 4 confirms these conclusions. [It appears that the graph of f( h) is the graph of f() shifted right if h is negative and left if h is positive the opposite of what ou might epect.] FIGURE 4 Horizontal shifts. 2 ( 3) 2 ( 2) 2 Matched Problem (A) How are the graphs of 3 and 1 related to the graph of? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How are the graphs of 3 and 1 related to the graph of? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. Comparing the graph of f() k with the graph of f(), we see that the graph of f() k can be obtained from the graph of f() b verticall translating (shifting) the graph of the latter upward k units if k is positive and downward k units if k is negative. Comparing the graph of f( h) with the graph of f(), we see that the graph of f( h) can be obtained from the graph of f() b horizontall translating (shifting) the graph of the latter h units to the left if h is positive and units to the right if h is negative. h EXAMPLE 6 Vertical and Horizontal Translations (Shifts) The graphs in Figure are either horizontal or vertical shifts of the graph of f(). Write appropriate equations for functions H, G, M, and N in terms of f.
9 2 Combining Functions 173 FIGURE Vertical and horizontal shifts. G f H M f N Solution Functions H and G are vertical shifts given b H() 3 G() 1 Functions M and N are horizontal shifts given b M() 2 N() 3 Matched Problem 6 The graphs in Figure 6 are either horizontal or vertical shifts of the graph of f() 3. Write appropriate equations for functions H, G, M, and N in terms of f. FIGURE 6 Vertical and horizontal shifts. G f H M f N Reflections, Epansions, and Contractions We now investigate how the graph of Af() is related to the graph of f() for different real numbers A. EXPLOREDISCUSS 3 (A) Graph A for A 1, 2, and 2 simultaneousl in the same coordinate sstem. (B) Graph A for A 1, 2, and 1 2 simultaneousl in the same coordinate sstem. (C) Describe the relationship between the graph of h() and the graph of G() A for A an real number. 1
10 174 2 Graphs and Functions Comparing the graph of Af() with the graph of f(), we see that the graph of Af() can be obtained from the graph of f() b multipling each ordinate value of the latter b A. The result is a vertical epansion of the graph of f() if A 1, a vertical contraction of the graph of f() if 0 A 1, and a reflection in the ais if A 1. EXAMPLE 7 Reflections, Epansions, and Contractions (A) How are the graphs of 2 3 and 0. 3 related to the graph of 3? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How is the graph of 2 3 related to the graph of 3? Confirm our answer b graphing both functions simultaneousl in the same coordinate sstem. Solution (A) The graph of 2 3 is a vertical epansion of the graph of 3 b a factor of 2, and the graph of 0. 3 is a vertical contraction of the graph of 3 b a factor of 0.. Figure 7 confirms this conclusion. FIGURE 7 Vertical epansion and contraction (B) The graph of 2 3 is a reflection in the ais and a vertical epansion of the graph of 3. Figure 8 confirms this conclusion. FIGURE 8 Reflection and vertical epansion Matched Problem 7 (A) How are the graphs of 2 and 0. related to the graph of? Confirm our answer b graphing all three functions simultaneousl in the same coordinate sstem. (B) How is the graph of 0. related to the graph of? Confirm our answer b graphing both functions in the same coordinate sstem.
11 2 Combining Functions 17 The various transformations considered above are summarized in the following bo for eas reference: Graph Transformations (Summar) Vertical Translation [see Fig. 9(a)]: k 0 Shift graph of f() up k units f() k k 0 Horizontal Translation [see Fig. 9(b)]: Reflection [see Fig. 9(c)]: Shift graph of f() down k units h 0 Shift graph of f() left h units f( h) h 0 Shift graph of f() right h units f() Reflect graph of f() in the ais Vertical Epansion and Contraction [see Fig. 9(d)]: Af() A 1 0 A 1 Verticall epand graph of f() b multipling each ordinate value b A Verticall contract graph of f() b multipling each ordinate value b A FIGURE 9 Graph transformations. g f h g f h f g f h g g() f () 2 g() f ( 3) g() f () g() 2f () h() f () 3 h() f ( 2) h() 0.f () (a) Vertical translation (b) Horizontal translation (c) Reflection (d) Epansion and contraction EXPLOREDISCUSS 4 Use a graphing utilit to eplore the graph of A( h) 2 k for various values of the constants A, h, and k. Discuss how the graph of A( h) 2 k is related to the graph of 2.
12 176 2 Graphs and Functions EXAMPLE 8 Combining Graph Transformations The graph of g() in Figure 10 is a transformation of the graph of 2. Find an equation for the function g. FIGURE 10 g() Solution To transform the graph of 2 [Fig. 11(a)] into the graph of g(), we first reflect the graph of 2 in the ais [Fig. 11(b)], then shift it to the right two units [Fig. 11(c)]. Thus, an equation for the function g is g() ( 2) 2 FIGURE ( 2) ( 2) 2 (a) (b) (c) FIGURE 12 Matched Problem 8 The graph of h() in Figure 12 is a transformation of the graph of 3. Find an equation for the function h. h()
13 2 Combining Functions 177 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. (A) The graph of 3 is the same as the graph of shifted upward 3 units, and the graph of 1 is the same as the graph of shifted downward 1 unit. The figure confirms these conclusions. (B) The graph of 3 is the same as the graph of shifted to the left 3 units, and the graph of 1 is the same as the graph of shifted to the right 1 unit. The figure confirms these conclusions G() ( 3) 3, H() ( 1) 3, M() 3 3, N() (A) The graph of 2 is a vertical epansion of the graph of, and the graph of 0. is a vertical contraction of the graph of. The figure confirms these conclusions. (B) The graph of 0. is a vertical contraction and a reflection in the ais of the graph of. The figure confirms this conclusion The graph of function h is a reflection in the ais and a horizontal translation of 3 units to the left of the graph of 3. An equation for h is h() ( 3) 3. EXERCISE 2 A Without looking back in the tet, indicate the domain and range of each of the following functions. (Making rough sketches on scratch paper ma help.) 1. h() 2. m() 3 3. g() f() 0.. F() G() 4 3 In Problems 7 10, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains. 7. f() 4; g() 1 8. f() 3; g() 2 9. f() 2 2 ; g() 2 1
14 178 2 Graphs and Functions 10. f() 3; g() 2 4 In Problems 11 14, for the indicated functions f and g, find the functions f g and g f, and find their domains. 11. f() 2 3; g() f() 2 ; g() f() 2 2/3 ; g() f() 4 3 ; g() 3 1/3 Problems 1 22 refer to the functions f and g given b the graphs below (the domain of each function is [ 2, 2]). Use the graph of f or g, as required, to graph each given function. f() g() In Problems 3 40, for the indicated functions f and g, find the functions f g and g f, and find their domains. 3. f() 2; g() f() 1; g() f() 3; g() 1 2 f() f() ; g() 2 4 f() 2 ; g() 3 4 ; g() 3 Each graph in Problems is the result of appling a sequence of transformations to the graph of one of the si basic functions in Figure 2. Identif the basic function and describe the transformation verball. Write an equation for the given graph. Check our equations in Problems b graphing each on a graphing utilit f() g() g( 2) 18. f( 1) 19. f() 20. g() g() f() B In Problems 23 28, indicate how the graph of each function is related to the graph of one of the si basic functions in Figure 2. Sketch a graph of each function. 42. Check our descriptions and graphs in Problems b graphing each function on a graphing utilit. 23. g() h() 4 2. f() ( 2) m() ( 1) f() g() In Problems 29 34, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains. 29. f() 2; g() f() ; g() f() 2 ; g() 3 f() 3 6; g() f() 2 2; g() f() ; g() 2 12
15 2 Combining Functions Changing the order in a sequence of transformations ma change the final result. Investigate each pair of transformations in Problems 3 8 to determine whether reversing their order can produce a different result. Support our conclusions with specific eamples and/or mathematical arguments. 3. Vertical shift, horizontal shift. 4. Vertical shift, reflection in ais.. Vertical shift, reflection in ais Vertical shift, vertical epansion. 7. Horizontal shift, reflection in ais. 8. Horizontal shift, vertical contraction. In Problems 9 66, 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. 9. h() (2 7) h() (3 ) h() h() h() h() h() h() 2 1 C In Problems 47 2, the graph of the function g is formed b appling the indicated sequence of transformations to the given function f. Find an equation for the function g and graph g using and. 47. The graph of f() is shifted 2 units to the left and 3 units up. 48. The graph of f() 3 is shifted 3 units to the right and 2 units down. 49. The graph of f() is reflected in the ais and shifted to the right 3 units. 0. The graph of f() is reflected in the ais and shifted to the left 1 unit. 1. The graph of f() 3 is reflected in the ais and shifted to the left 2 units and up 1 unit. 2. The graph of f() 2 is reflected in the ais and shifted to the right 2 units and down 4 units. Each of the following graphs involves a reflection in the ais and/or a vertical epansion or contraction of one of the basic functions in Figure 2. Identif the basic function and describe the transformation verball. Write an equation for the given graph. Check our equations in Problems b graphing each on a graphing utilit. 67.
16 180 2 Graphs and Functions 68. In Problems 79 84, for the indicated functions f and g, find the functions f g and g f, and find their domains. 79. f() 2 ; g() f() 16; g() 2 f() ; g() f() 2 ; g() f() 2 ; g() 2 4 f() 2 8; g() 2 9 APPLICATIONS 8. Market Research. The demand and the price p (in dollars) for a certain product are related b 70. f( p) 4, p The revenue (in dollars) from the sale of units is given b R() and the cost (in dollars) of producing units is given b 71. Are the functions fg and gf identical? Justif our answer. 72. Are the functions f g and g f identical? Justif our answer. 73. Is there a function g that satisfies f g g f f for all functions f? If so, what is it? 74. Is there a function g that satisfies fg gf f for all functions f? If so, what is it? In Problems 7 78, for the indicated functions f and g, find the functions f g, f g, fg, and f/g, and find their domains f() 1 ; f() 1; g() 6 1 f() 1 ; g() 1 g() f() ; g() C() 10 30,000 Epress the profit as a function of the price p. 86. Market Research. The demand and the price p (in dollars) for a certain product are related b f( p), p The revenue (in dollars) from the sale of units and the cost (in dollars) of producing units are given, respectivel, b 1 R() 0 and C() 20 40, Epress the profit as a function of the price p. 87. Famil of Curves. In calculus, solutions to certain tpes of problems often involve an unspecified constant. For eample, consider the equation 1 C 2 C where C is a positive constant. The collection of graphs of this equation for all permissible values of C is called a famil of curves. On the same aes, graph the members of this famil corresponding to C 1, 2, 3, and 4.
17 2 Combining Functions Famil of Curves. A famil of curves is defined b the equation 2C 2 C 2 where C is a positive constant. On the same aes, graph the members of this famil corresponding to C 1, 2, 3, and Fluid Flow. A cubic tank is 4 feet on a side and is initiall full of water. Water flows out an opening in the bottom of the tank at a rate proportional to the square root of the depth (see the figure). Using advanced concepts from mathematics and phsics, it can be shown that the volume of the water in the tank t minutes after the water begins to flow is given b V(t) 64 (C t)2 0 2 C t C where C is a constant that depends on the size of the opening. Graph V(t) for C 1, C 2, C 4, and C 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. 4 inches h r 1 V r 2 h 3 4 inches 4 feet 4 feet (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 4 feet h(t) 0. t epress V as a function of t. 90. Evaporation. A water trough with triangular ends is 9 feet long, 4 feet wide, and 2 feet deep (see the figure). Initiall, the trough is full of water, but due to evaporation, the volume of the water in the trough decreases at a rate proportional to the square root of the volume. Using advanced concepts from mathematics and phsics, it can be shown that the volume after t hours is given b where C is a constant. Graph V(t) for C 4, C, and C 6. 2 feet V(t) 1 (t 6C)2 0 2 C 9 feet t 6 C 4 feet 92. 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. 2 feet 6 feet (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) t epress V as a function of t. h w V 3wh 4 feet
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