Functions and Their Graphs PDF

333371_0106.qxd 12/27/06 10:32 AM Page 136 136 Chapter 1 Functions and Their Graphs 1.6 Combinations of Functions Arithmetic Combinations of Functions What you should learn...

333371_0106.qxd 12/27/06 10:32 AM Page 136 136 Chapter 1 Functions and Their Graphs 1.6 Combinations of Functions Arithmetic Combinations of Functions What you should learn 䊏 Add, subtract, multiply, and divide Just as two real numbers can be combined by the operations of addition, functions. subtraction, multiplication, and division to form other real numbers, two 䊏 Find compositions of one function with functions can be combined to create new functions. If f 共x兲 ⫽ 2x ⫺ 3 and another function. g共x兲 ⫽ x 2 ⫺ 1, you can form the sum, difference, product, and quotient of f and 䊏 Use combinations of functions to model g as follows. and solve real-life problems. f 共x兲 ⫹ g共x兲 ⫽ 共2x ⫺ 3兲 ⫹ 共x 2 ⫺ 1兲 Why you should learn it Combining functions can sometimes help ⫽ x 2 ⫹ 2x ⫺ 4 Sum you better understand the big picture. For f 共x兲 ⫺ g共x兲 ⫽ 共2x ⫺ 3兲 ⫺ 共x 2 ⫺ 1兲 instance, Exercises 75 and 76 on page 145 illustrate how to use combinations of func- ⫽ ⫺x 2 ⫹ 2x ⫺ 2 Difference tions to analyze U.S. health care expenditures. f 共x兲 ⭈ g共x兲 ⫽ 共2x ⫺ 3兲共x 2 ⫺ 1兲 ⫽ 2x 3 ⫺ 3x 2 ⫺ 2x ⫹ 3 Product f 共x兲 2x ⫺ 3 ⫽ 2 , x ⫽ ±1 Quotient g共x兲 x ⫺1 The domain of an arithmetic combination of functions f and g consists of all real numbers that are common to the domains of f and g. In the case of the quotient f 共x兲兾g共x兲, there is the further restriction that g共x兲 ⫽ 0. SuperStock Sum, Difference, Product, and Quotient of Functions Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows. 1. Sum: 共 f ⫹ g兲共x兲 ⫽ f 共x兲 ⫹ g共x兲 2. Difference: 共 f ⫺ g兲共x兲 ⫽ f 共x兲 ⫺ g共x兲 3. Product: 共 fg兲共x兲 ⫽ f 共x兲 ⭈ g共x兲 f 共x兲 冢g 冣共x兲 ⫽ g共x兲, f 4. Quotient: g共x兲 ⫽ 0 Example 1 Finding the Sum of Two Functions Given f 共x兲 ⫽ 2x ⫹ 1 and g共x兲 ⫽ x 2 ⫹ 2x ⫺ 1, find 共 f ⫹ g兲共x兲. Then evaluate the sum when x ⫽ 2. Solution 共 f ⫹ g兲共x兲 ⫽ f 共x兲 ⫹ g共x兲 ⫽ 共2x ⫹ 1兲 ⫹ 共x 2 ⫹ 2x ⫺ 1兲 ⫽ x2 ⫹ 4x When x ⫽ 2, the value of this sum is 共 f ⫹ g兲共2兲 ⫽ 22 ⫹ 4共2兲 ⫽ 12. Now try Exercise 7(a). 333371_0106.qxd 12/27/06 10:32 AM Page 137 Section 1.6 Combinations of Functions 137 Example 2 Finding the Difference of Two Functions Given f 共x兲 ⫽ 2x ⫹ 1 and g共x兲 ⫽ x 2 ⫹ 2x ⫺ 1, find 共 f ⫺ g兲共x兲. Then evaluate the difference when x ⫽ 2. Algebraic Solution Graphical Solution The difference of the functions f and g is You can use a graphing utility to graph the difference of two functions. Enter the functions as follows (see Figure 1.70). 共 f ⫺ g兲共x兲 ⫽ f 共x兲 ⫺ g共x兲 y1 ⫽ 2x ⫹ 1 ⫽ 共2x ⫹ 1兲 ⫺ 共x 2 ⫹ 2x ⫺ 1兲 y2 ⫽ x2 ⫹ 2x ⫺ 1 ⫽ ⫺x 2 ⫹ 2. y3 ⫽ y1 ⫺ y2 When x ⫽ 2, the value of this difference is Graph y3 as shown in Figure 1.71. Then use the value feature 共 f ⫺ g兲共2兲 ⫽ ⫺ 共2兲 2 ⫹ 2 or the zoom and trace features to estimate that the value of the ⫽ ⫺2. difference when x ⫽ 2 is ⫺2. Note that 共 f ⫺ g兲共2兲 can also be evaluated as follows. y3 = −x2 + 2 3 共 f ⫺ g兲共2兲 ⫽ f 共2兲 ⫺ g共2兲 ⫽ 关2共2兲 ⫹ 1兴 ⫺ 关22 ⫹ 2共2兲 ⫺ 1兴 −5 4 ⫽5⫺7 ⫽ ⫺2 −3 Figure 1.70 Figure 1.71 Now try Exercise 7(b). In Examples 1 and 2, both f and g have domains that consist of all real numbers. So, the domain of both 共 f ⫹ g兲 and 共 f ⫺ g兲 is also the set of all real numbers. Remember that any restrictions on the domains of f or g must be considered when forming the sum, difference, product, or quotient of f and g. For instance, the domain of f 共x兲 ⫽ 1兾x is all x ⫽ 0, and the domain of g共x兲 ⫽ 冪x is 关0, ⬁兲. This implies that the domain of 共 f ⫹ g兲 is 共0, ⬁兲. Example 3 Finding the Product of Two Functions Additional Examples a. Find 共 fg 兲共x兲 given that f 共x兲 ⫽ x ⫹ 5 and Given f 共x兲 ⫽ x2 and g共x兲 ⫽ x ⫺ 3, find 共 fg兲共x兲. Then evaluate the product g共x兲 ⫽ 3x. when x ⫽ 4. Solution 共 fg兲共x兲 ⫽ f 共x兲 ⭈ g共x兲 Solution ⫽ 共x ⫹ 5兲共3x兲 ⫽ 3x 2 ⫹ 15x 共 fg兲共x兲 ⫽ f 共x兲g 共x兲 1 b. Find 共g f 兲共x兲 given that f 共x兲 ⫽ and x ⫽ 共x 2兲共x ⫺ 3兲 g共x兲 ⫽ x. x⫹1 ⫽ x3 ⫺ 3x 2 Solution 共gf 兲 共x兲 ⫽ g共x兲 ⭈ f 共x兲 When x ⫽ 4, the value of this product is 共 fg兲共4兲 ⫽ 43 ⫺ 3共4兲2 ⫽ 16. ⫽冢 x x⫹1 x冣冢 冣1 1 ⫽ , x⫽0 Now try Exercise 7(c). x⫹1 333371_0106.qxd 12/27/06 10:32 AM Page 138 138 Chapter 1 Functions and Their Graphs Example 4 Finding the Quotient of Two Functions Find 共 f兾g兲共x兲 and 共g兾f 兲共x兲 for the functions given by f 共x兲 ⫽ 冪x and g共x兲 ⫽ 冪4 ⫺ x2. Then find the domains of f兾g and g兾f. Solution The quotient of f and g is f 共x兲 冪x 冢g 冣共x兲 ⫽ g共x兲 ⫽ 冪4 ⫺ x , f 2 and the quotient of g and f is g共x兲 冪4 ⫺ x 2 冢冣 g 共x兲 ⫽ ⫽ f f 共x兲 冪x. 5 y3 = ( gf ((x) = x 4 − x2 The domain of f is 关0, ⬁兲 and the domain of g is 关⫺2, 2兴. The intersection of these domains is 关0, 2兴. So, the domains for f兾g and g兾f are as follows. Domain of 共 f兾g兲: 关0, 2兲 Domain of 共g兾f 兲: 共0, 2兴 −3 6 Now try Exercise 7(d). −1 Figure 1.72 TECHNOLOGY TIP You can confirm the domain of f兾g in Example 4 with your graphing utility by entering the three functions y1 ⫽ 冪x, y2 ⫽ 冪4 ⫺ x2, and y3 ⫽ y1兾y2, and graphing y3, as shown in Figure 1.72. Use the trace feature 4 − x2 to determine that the x-coordinates of points on the graph extend from 0 to 2 y4 = ( gf ((x) = x but do not include 2. So, you can estimate the domain of f兾g to be 关0, 2兲. 5 You can confirm the domain of g兾f in Example 4 by entering y4 ⫽ y2兾y1 and graphing y4 , as shown in Figure 1.73. Use the trace feature to determine that the x-coordinates of points on the graph extend from 0 to 2 but do not include 0. So, you can estimate the domain of g兾f to be 共0, 2兴. −3 6 −1 Compositions of Functions Figure 1.73 Another way of combining two functions is to form the composition of one with the other. For instance, if f 共x兲 ⫽ x 2 and g共x兲 ⫽ x ⫹ 1, the composition of f with g is f 共g共x兲兲 ⫽ f 共x ⫹ 1兲 ⫽ 共x ⫹ 1兲2. This composition is denoted as f ⬚ g and is read as “f composed with g.” f °g Definition of Composition of Two Functions The composition of the function f with the function g is x g(x) f f(g(x)) 共 f ⬚ g兲共x兲 ⫽ f 共 g共x兲兲. g Domain of g The domain of f ⬚ g is the set of all x in the domain of g such that g共x兲 is in Domain of f the domain of f. (See Figure 1.74.) Figure 1.74 333371_0106.qxd 12/27/06 10:32 AM Page 139 Section 1.6 Combinations of Functions 139 Example 5 Forming the Composition of f with g Find 共 f ⬚ g兲共x兲 for f 共x兲 ⫽ 冪x, x ≥ 0, and g共x兲 ⫽ x ⫺ 1, x ≥ 1. If possible, find 共 f ⬚ g兲共2兲 and 共 f ⬚ g兲共0兲. Exploration Let f 共x兲 ⫽ x ⫹ 2 and Solution g共x兲 ⫽ 4 ⫺ x 2. Are the compo- 共 f ⬚ g兲共x兲 ⫽ f 共 g共x兲兲 Definition of f ⬚ g sitions f ⬚ g and g ⬚ f equal? You can use your graphing utility to ⫽ f 共x ⫺ 1兲 Definition of g共x兲 answer this question by entering ⫽ 冪x ⫺ 1, x ≥ 1 Definition of f 共x兲 and graphing the following functions. The domain of f ⬚ g is 关1, ⬁兲. So, 共 f ⬚ g兲共2兲 ⫽ 冪2 ⫺ 1 ⫽ 1 is defined, but 共 f ⬚ g兲共0兲 is not defined because 0 is not in the domain of f ⬚ g. y1 ⫽ 共4 ⫺ x 2兲 ⫹ 2 Now try Exercise 35. y2 ⫽ 4 ⫺ 共x ⫹ 2兲2 What do you observe? Which The composition of f with g is generally not the same as the composition of function represents f ⬚ g and g with f. This is illustrated in Example 6. which represents g ⬚ f ? Example 6 Compositions of Functions Given f 共x兲 ⫽ x ⫹ 2 and g共x兲 ⫽ 4 ⫺ x2, evaluate (a) 共 f ⬚ g兲共x兲 and (b) 共g ⬚ f 兲共x兲 when x ⫽ 0, 1, 2, and 3. Algebraic Solution Numerical Solution a. 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 Definition of f ⬚ g a. You can use the table feature of a graphing utility to ⫽ f 共4 ⫺ x 2兲 Definition of g共x兲 evaluate f ⬚ g when x ⫽ 0, 1, 2, and 3. Enter y1 ⫽ g共x兲 and y2 ⫽ f 共g共x兲兲 in the equation editor (see Figure ⫽ 共4 ⫺ x 2兲 ⫹ 2 Definition of f 共x兲 1.75). Then set the table to ask mode to find the desired ⫽ ⫺x ⫹ 6 2 function values (see Figure 1.76). Finally, display the 共f ⬚ g 兲共0 兲 ⫽ ⫺02 ⫹ 6 ⫽6 table, as shown in Figure 1.77. 共f ⬚ g兲共1兲 ⫽ ⫺1 ⫹ 6 2 ⫽5 b. You can evaluate g ⬚ f when x ⫽ 0, 1, 2, and 3 by using 共f ⬚ g兲共2兲 ⫽ ⫺2 ⫹ 6 2 ⫽2 a procedure similar to that of part (a). You should obtain 共f ⬚ g兲共3兲 ⫽ ⫺3 ⫹ 6 2 ⫽ ⫺3 the table shown in Figure 1.78. b. 共g ⬚ f 兲共x兲 ⫽ g共 f (x)兲 Definition of g ⬚ f ⫽ g共x ⫹ 2兲 Definition of f 共x兲 ⫽ 4 ⫺ 共x ⫹ 2兲2 Definition of g共x兲 ⫽ 4 ⫺ 共x 2 ⫹ 4x ⫹ 4兲 ⫽ ⫺x 2 ⫺ 4x Figure 1.75 Figure 1.76 共g ⬚ f 兲共0兲 ⫽ ⫺02 ⫺ 4共0兲 ⫽ 0 共g ⬚ f 兲共1兲 ⫽ ⫺12 ⫺ 4共1兲 ⫽ ⫺5 共g ⬚ f 兲共2兲 ⫽ ⫺22 ⫺ 4共2兲 ⫽ ⫺12 共g ⬚ f 兲共3兲 ⫽ ⫺32 ⫺ 4共3兲 ⫽ ⫺21 Note that f ⬚ g ⫽ g ⬚ f. Figure 1.77 Figure 1.78 Now try Exercise 37. From the tables you can see that f ⬚ g ⫽ g ⬚ f. 333371_0106.qxd 12/27/06 10:33 AM Page 140 140 Chapter 1 Functions and Their Graphs To determine the domain of a composite function f ⬚ g, you need to restrict the outputs of g so that they are in the domain of f. For instance, to find the domain of f ⬚ g given that f 共x兲 ⫽ 1兾x and g共x兲 ⫽ x ⫹ 1, consider the outputs of g. These can be any real number. However, the domain of f is restricted to all real numbers except 0. So, the outputs of g must be restricted to all real numbers except 0. This means that g共x兲 ⫽ 0, or x ⫽ ⫺1. So, the domain of f ⬚ g is all real numbers except x ⫽ ⫺1. Example 7 Finding the Domain of a Composite Function Find the domain of the composition 共 f ⬚ g兲共x兲 for the functions given by f 共x兲 ⫽ x 2 ⫺ 9 and g共x兲 ⫽ 冪9 ⫺ x 2. Algebraic Solution Graphical Solution The composition of the functions is as follows. You can use a graphing utility to graph the composition of the functions 共 f ⬚ g兲共x兲 as y ⫽ 共冪9 ⫺ x2 兲 ⫺ 9. Enter the functions as follows. 2 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 ⫽ f 共冪9 ⫺ x2 兲 y1 ⫽ 冪9 ⫺ x2 y2 ⫽ y12 ⫺ 9 ⫽ 共冪9 ⫺ x2 兲 ⫺ 9 2 Graph y2 , as shown in Figure 1.79. Use the trace feature to determine that the x-coordinates of points on the graph extend from ⫺3 to 3. So, ⫽ 9 ⫺ x2 ⫺ 9 you can graphically estimate the domain of 共 f ⬚ g兲共x兲 to be 关⫺3, 3兴. ⫽ ⫺x 2 0 −4 4 From this, it might appear that the domain of 2 the composition is the set of all real numbers. y= ( 9 − x2 ( − 9 This, however, is not true. Because the domain of f is the set of all real numbers and the domain of g is 关⫺3, 3兴, the domain of 共 f ⬚ g兲 is −12 关⫺3, 3兴. Figure 1.79 Now try Exercise 39. Example 8 A Case in Which f ⬚ g ⫽ g ⬚ f Given f 共x兲 ⫽ 2x ⫹ 3 and g共x兲 ⫽ 12共x ⫺ 3兲, find each composition. a. 共 f ⬚ g兲共x兲 b. 共g ⬚ f 兲共x兲 STUDY TIP In Example 8, note that the Solution two composite functions f ⬚ g a. 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲 b. 共g ⬚ f 兲共x兲 ⫽ g共 f (x)兲 and g ⬚ f are equal, and both represent the identity function. ⫽ g共2x ⫹ 3兲 ⫽f 冢12 共x ⫺ 3兲冣 That is, 共 f ⬚ g兲共x兲 ⫽ x and 共g ⬚ f 兲共x兲 ⫽ x. You will study 冤 冥 1 ⫽ 共2x ⫹ 3兲 ⫺ 3 this special case in the next 冤 12 共x ⫺ 3兲冥 ⫹ 3 2 ⫽2 section. 1 ⫽ 共2x兲 ⫽x⫺3⫹3⫽x 2 ⫽x ⫽x Now try Exercise 51. 333371_0106.qxd 12/27/06 10:33 AM Page 141 Section 1.6 Combinations of Functions 141 In Examples 5, 6, 7, and 8, you formed the composition of two given func- tions. In calculus, it is also important to be able to identify two functions that Exploration make up a given composite function. Basically, to “decompose” a composite Write each function as a function, look for an “inner” and an “outer” function. composition of two functions. ⱍ a. h共x兲 ⫽ x3 ⫺ 2 ⱍ Example 9 Identifying a Composite Function ⱍ ⱍ b. r共x兲 ⫽ x3 ⫺ 2 Write the function h共x兲 ⫽ 共3x ⫺ 5兲3 as a composition of two functions. What do you notice about the inner and outer functions? Solution One way to write h as a composition of two functions is to take the inner func- tion to be g共x兲 ⫽ 3x ⫺ 5 and the outer function to be f 共x兲 ⫽ x3. Then you can Activities write 1. Find 共 f ⫹ g兲共⫺1兲 and 冢gf 冣共2兲 for h共x兲 ⫽ 共3x ⫺ 5兲3 f 共x兲 ⫽ 3x 2 ⫹ 2, g共x兲 ⫽ 2x. 7 ⫽ f 共3x ⫺ 5兲 Answer: 3; 2 2. Given f 共x兲 ⫽ 3x 2 ⫹ 2 and g共x兲 ⫽ 2x, ⫽ f 共g共x兲兲. find f ⬚ g. Now try Exercise 65. Answer: 共 f ⬚ g兲共x兲 ⫽ 12x 2 ⫹ 2 3. Find two functions f and g such that 共 f ⬚ g兲共x兲 ⫽ h共x兲. (There are many correct answers.) Example 10 Identifying a Composite Function 1 h共x兲 ⫽. 冪3x ⫹ 1 Write the function Answer: 1 1 f 共x兲 ⫽ and g共x兲 ⫽ 3x ⫹ 1 h共x兲 ⫽ 冪x 共x ⫺ 2兲 2 as a composition of two functions. Exploration Solution The function in Example 10 can One way to write h as a composition of two functions is to take the inner function be decomposed in other ways. to be g共x兲 ⫽ x ⫺ 2 and the outer function to be For which of the following pairs of functions is h共x兲 equal to 1 f 共x兲 ⫽ f 共g共x兲兲? x2 1 ⫽ x⫺2. a. g共x兲 ⫽ and x⫺2 Then you can write f 共x兲 ⫽ x 2 1 b. g共x兲 ⫽ x 2 and h共x兲 ⫽ 共x ⫺ 2兲2 1 f 共x兲 ⫽ ⫽ 共x ⫺ 2兲⫺2 x⫺2 ⫽ f 共x ⫺ 2兲 c. g共x兲 ⫽ 1 and x ⫽ f 共g共x兲兲. f 共x兲 ⫽ 共x ⫺ 2兲2 Now try Exercise 69. 333371_0106.qxd 12/27/06 10:33 AM Page 142 142 Chapter 1 Functions and Their Graphs Example 11 Bacteria Count Exploration The number N of bacteria in a refrigerated food is given by Use a graphing utility to graph N共T 兲 ⫽ 20T 2 ⫺ 80T ⫹ 500, 2 ≤ T ≤ 14 y1 ⫽ 320x 2 ⫹ 420 and y2 ⫽ 2000 in the same viewing where T is the temperature of the food (in degrees Celsius). When the food is window. (Use a viewing win- removed from refrigeration, the temperature of the food is given by dow in which 0 ≤ x ≤ 3 and T共t兲 ⫽ 4t ⫹ 2, 0 ≤ t ≤ 3 400 ≤ y ≤ 4000.) Explain how the graphs can be used to where t is the time (in hours). answer the question asked in a. Find the composition N共T共t兲兲 and interpret its meaning in context. Example 11(c). Compare your b. Find the number of bacteria in the food when t ⫽ 2 hours. answer with that given in part (c). When will the bacteria c. Find the time when the bacterial count reaches 2000. count reach 3200? Solution Notice that the model for this bacteria count situation is valid a. N共T共t兲兲 ⫽ 20共4t ⫹ 2兲2 ⫺ 80共4t ⫹ 2兲 ⫹ 500 only for a span of 3 hours. Now ⫽ 20共16t 2 ⫹ 16t ⫹ 4兲 ⫺ 320t ⫺ 160 ⫹ 500 suppose that the minimum num- ber of bacteria in the food is ⫽ 320t 2 ⫹ 320t ⫹ 80 ⫺ 320t ⫺ 160 ⫹ 500 reduced from 420 to 100. Will ⫽ 320t 2 ⫹ 420 the number of bacteria still reach a level of 2000 within the The composite function N共T共t兲兲 represents the number of bacteria as a function three-hour time span? Will the of the amount of time the food has been out of refrigeration. number of bacteria reach a level b. When t ⫽ 2, the number of bacteria is of 3200 within 3 hours? N ⫽ 320共2兲 2 ⫹ 420 ⫽ 1280 ⫹ 420 ⫽ 1700. c. The bacterial count will reach N ⫽ 2000 when 320t 2 ⫹ 420 ⫽ 2000. You can N = 320t2 + 420, 2 ≤ t ≤ 3 solve this equation for t algebraically as follows. 3500 320t 2 ⫹ 420 ⫽ 2000 320t 2 ⫽ 1580 79 t2 ⫽ 2 3 16 1500 冪79 Figure 1.80 t⫽ 4 t ⬇ 2.22 hours 2500 So, the count will reach 2000 when t ⬇ 2.22 hours. When you solve this equation, note that the negative value is rejected because it is not in the domain of the composite function. You can use a graphing utility to confirm your solution. First graph the equation N ⫽ 320t 2 ⫹ 420, as shown in Figure 1.80. 2 3 Then use the zoom and trace features to approximate N ⫽ 2000 when 1500 t ⬇ 2.22, as shown in Figure 1.81. Figure 1.81 Now try Exercise 81. 333371_0106.qxd 12/27/06 10:33 AM Page 143 Section 1.6 Combinations of Functions 143 1.6 Exercises See www.CalcChat.com for worked-out solutions to odd-numbered exercises. Vocabulary Check Fill in the blanks. 1. Two functions f and g can be combined by the arithmetic operations of _______ , _______ , _______ , and _______ to create new functions. 2. The _______ of the function f with the function g is 共 f ⬚ g兲共x兲 ⫽ f 共g共x兲兲. 3. The domain of f ⬚ g is the set of all x in the domain of g such that _______ is in the domain of f. 4. To decompose a composite function, look for an _______ and an _______ function. In Exercises 1– 4, use the graphs of f and g to graph In Exercises 13–26, evaluate the indicated function for h冇x冈 ⴝ 冇 f 1 g冈冇x冈. To print an enlarged copy of the graph, f 冇x冈 ⴝ x2 ⴚ 1 and g冇x冈 ⴝ x ⴚ 2 algebraically. If possible, go to the website www.mathgraphs.com. use a graphing utility to verify your answer. 1. y 2. y 13. 共 f ⫹ g兲共3兲 14. 共 f ⫺ g兲共⫺2兲 3 f 3 g 15. 共 f ⫺ g兲共0兲 16. 共 f ⫹ g兲共1兲 2 2 1 17. 共 fg兲共4兲 18. 共 f g兲共⫺6兲 g f x x 冢g 冣共⫺5兲 冢g 冣共0兲 f f −2 −1 1 2 3 4 −3 −2 −1 2 3 19. 20. −2 −2 −3 −3 21. 共 f ⫺ g兲共2t兲 22. 共 f ⫹ g兲共t ⫺ 4兲 23. 共 fg兲共⫺5t兲 24. 共 fg兲共3t2兲 3. y 4. y 冢g 冣共⫺t兲 冢gf 冣共t ⫹ 2兲 f 5 3 25. 26. 4 g f f 1 2 x In Exercises 27–30, use a graphing utility to graph the func- g −3 −2 1 3 tions f, g, and h in the same viewing window. −1 x −2 27. f 共x兲 ⫽ 12 x, g共x兲 ⫽ x ⫺ 1, h共x兲 ⫽ f 共x兲 ⫹ g共x兲 −2 −1 1 2 3 4 −3 28. f 共x兲 ⫽ 13 x, g共x兲 ⫽ ⫺x ⫹ 4, h共x兲 ⫽ f 共x兲 ⫺ g共x兲 In Exercises 5 –12, find (a) 冇 f 1 g冈冇x冈, (b) 冇 f ⴚ g冈冇x冈, 29. f 共x兲 ⫽ x , 2 g共x兲 ⫽ ⫺2x, h共x兲 ⫽ f 共x兲 ⭈ g共x兲 (c) 冇 fg冈冇x冈, and (d) 冇 f/g冈冇x冈. What is the domain of f/g? 30. f 共x兲 ⫽ 4 ⫺ x 2, g共x兲 ⫽ x, h共x兲 ⫽ f 共x兲兾g共x兲 5. f 共x兲 ⫽ x ⫹ 3, g共x兲 ⫽ x ⫺ 3 6. f 共x兲 ⫽ 2x ⫺ 5, g共x兲 ⫽ 1 ⫺ x In Exercises 31–34, use a graphing utility to graph f, g, and f 1 g in the same viewing window. Which function con- 7. f 共x兲 ⫽ x 2, g共x兲 ⫽ 1 ⫺ x tributes most to the magnitude of the sum when 0 } x } 2? 8. f 共x兲 ⫽ 2x ⫺ 5, g共x兲 ⫽ 4 Which function contributes most to the magnitude of the sum when x > 6? 9. f 共x兲 ⫽ x 2 ⫹ 5, g共x兲 ⫽ 冪1 ⫺ x x3 x2 31. f 共x兲 ⫽ 3x, g共x兲 ⫽ ⫺ 10. f 共x兲 ⫽ 冪x 2 ⫺ 4, g共x兲 ⫽ 10 x ⫹1 2 x 1 1 32. f 共x兲 ⫽ , g共x兲 ⫽ 冪x 11. f 共x兲 ⫽ , g共x兲 ⫽ 2 2 x x 33. f 共x兲 ⫽ 3x ⫹ 2, g共x兲 ⫽ ⫺ 冪x ⫹ 5 x 12. f 共x兲 ⫽ , g共x兲 ⫽ x 3 x⫹1 34. f 共x兲 ⫽ x2 ⫺ 12, g共x兲 ⫽ ⫺3x2 ⫺ 1 333371_0106.qxd 12/27/06 10:33 AM Page 144 144 Chapter 1 Functions and Their Graphs In Exercises 35–38, find (a) f ⬚ g, (b) g ⬚ f, and, if possible, ⱍⱍ 59. f 共x兲 ⫽ x , g共x兲 ⫽ 2x3 (c) 冇 f ⬚ g冈冇0冈. 6 60. f 共x兲 ⫽ , g共x兲 ⫽ ⫺x 35. f 共x兲 ⫽ x2, g共x兲 ⫽ x ⫺ 1 3x ⫺ 5 36. f 共x兲 ⫽ 冪 3 x ⫺ 1, g共x兲 ⫽ x 3 ⫹ 1 In Exercises 61–64, use the graphs of f and g to evaluate the 37. f 共x兲 ⫽ 3x ⫹ 5, g共x兲 ⫽ 5 ⫺ x functions. 1 38. f 共x兲 ⫽ x 3, g共x兲 ⫽ y x y y = f (x ) 4 y = g (x ) 4 In Exercises 39–48, determine the domains of (a) f, 3 (b) g, and (c) f ⬚ g. Use a graphing utility to verify your 3 results. 2 2 1 39. f 共x兲 ⫽ 冪x ⫹ 4 , g共x兲 ⫽ x2 1 x x x 1 2 3 4 40. f 共x兲 ⫽ 冪x ⫹ 3, g 共x兲 ⫽ 1 2 3 4 2 41. f 共x兲 ⫽ x2 ⫹ 1 , g共x兲 ⫽ 冪x 61. (a) 共 f ⫹ g兲共3兲 (b) 共 f兾g兲共2兲 42. f 共x兲 ⫽ x1兾4 , g共x兲 ⫽ x4 62. (a) 共 f ⫺ g兲共1兲 (b) 共 f g兲共4兲 1 63. (a) 共 f ⬚ g兲共2兲 (b) 共g ⬚ f 兲共2兲 43. f 共x兲 ⫽ , g共x兲 ⫽ x ⫹ 3 x 64. (a) 共 f ⬚ g兲共1兲 (b) 共g ⬚ f 兲共3兲 1 1 44. f 共x兲 ⫽ , g共x兲 ⫽ In Exercises 65–72, find two functions f and g such that x 2x 冇 f ⬚ g冈冇x冈 ⴝ h冇x冈. (There are many correct answers.) ⱍ ⱍ 45. f 共x兲 ⫽ x ⫺ 4 , g共x兲 ⫽ 3 ⫺ x 2 65. h共x兲 ⫽ 共2x ⫹ 1兲2 66. h共x兲 ⫽ 共1 ⫺ x兲3 46. f 共x兲 ⫽ , g共x兲 ⫽ x ⫺ 1 x ⱍⱍ 67. h共x兲 ⫽ 冪 3 x2 ⫺4 68. h共x兲 ⫽ 冪9 ⫺ x 1 1 47. f 共x兲 ⫽ x ⫹ 2 , g共x兲 ⫽ 69. h共x兲 ⫽ x2 ⫺ 4 x⫹2 3 4 48. f 共x兲 ⫽ , g共x兲 ⫽ x ⫹ 1 70. h共x兲 ⫽ x2 ⫺1 共5x ⫹ 2兲2 71. h共x兲 ⫽ 共x ⫹ 4兲 2 ⫹ 2共x ⫹ 4兲 In Exercises 49– 54, (a) find f ⬚ g, g ⬚ f, and the domain of 72. h共x兲 ⫽ 共x ⫹ 3兲3兾2 ⫹ 4共x ⫹ 3兲1兾2 f ⬚ g. (b) Use a graphing utility to graph f ⬚ g and g ⬚ f. Determine whether f ⬚ g ⴝ g ⬚ f. 73. Stopping Distance The research and development 49. f 共x兲 ⫽ 冪x ⫹ 4, g共x兲 ⫽ x2 department of an automobile manufacturer has determined that when required to stop quickly to avoid an accident, the 50. f 共x兲 ⫽ 冪 3 x ⫹ 1, g共x兲 ⫽ x 3 ⫺ 1 distance (in feet) a car travels during the driver’s reaction 51. f 共x兲 ⫽ 13 x ⫺ 3, g共x兲 ⫽ 3x ⫹ 9 time is given by 52. f 共x兲 ⫽ 冪x , g共x兲 ⫽ 冪x R共x兲 ⫽ 4 x 3 53. f 共x兲 ⫽ x 2兾3, g共x兲 ⫽ x6 ⱍⱍ 54. f 共x兲 ⫽ x , g共x兲 ⫽ ⫺x2 ⫹ 1 where x is the speed of the car in miles per hour. The distance (in feet) traveled while the driver is braking is given by In Exercises 55–60, (a) find 冇 f ⬚ g冈冇x冈 and 冇 g ⬚ f 冈冇x冈, (b) determine algebraically whether 冇 f ⬚ g冈冇x冈 ⴝ 冇 g ⬚ f 冈冇x冈, and B共x兲 ⫽ 15 1 2 x. (c) use a graphing utility to complete a table of values for the two compositions to confirm your answers to part (b). (a) Find the function that represents the total stopping distance T. 55. f 共x兲 ⫽ 5x ⫹ 4, g共x兲 ⫽ 4 ⫺ x (b) Use a graphing utility to graph the functions R, B, and 56. f 共x兲 ⫽ 14共x ⫺ 1兲, g共x兲 ⫽ 4x ⫹ 1 T in the same viewing window for 0 ≤ x ≤ 60. 57. f 共x兲 ⫽ 冪x ⫹ 6, g共x兲 ⫽ x2 ⫺ 5 (c) Which function contributes most to the magnitude of 58. f 共x兲 ⫽ x3 ⫺ 4, g共x兲 ⫽ 冪 3 x ⫹ 10 the sum at higher speeds? Explain. 333371_0106.qxd 12/27/06 10:34 AM Page 145 Section 1.6 Combinations of Functions 145 74. Sales From 2000 to 2006, the sales R1 (in thousands of 78. Geometry A square concrete foundation was prepared as dollars) for one of two restaurants owned by the same par- a base for a large cylindrical gasoline tank (see figure). ent company can be modeled by R1 ⫽ 480 ⫺ 8t ⫺ 0.8t 2, for t ⫽ 0, 1, 2, 3, 4, 5, 6, where t ⫽ 0 represents 2000. During the same seven-year period, the sales R 2 (in thou- sands of dollars) for the second restaurant can be modeled by R2 ⫽ 254 ⫹ 0.78t, for t ⫽ 0, 1, 2, 3, 4, 5, 6. r (a) Write a function R3 that represents the total sales for the two restaurants. (b) Use a graphing utility to graph R1, R 2, and R3 (the total sales function) in the same viewing window. x Data Analysis In Exercises 75 and 76, use the table, which (a) Write the radius r of the tank as a function of the length shows the total amounts spent (in billions of dollars) on x of the sides of the square. health services and supplies in the United States and Puerto (b) Write the area A of the circular base of the tank as a Rico for the years 1995 through 2005. The variables y1, y2, function of the radius r. and y3 represent out-of-pocket payments, insurance premi- (c) Find and interpret 共A ⬚ r兲共x兲. ums, and other types of payments, respectively. (Source: U.S. Centers for Medicare and Medicaid Services) 79. Cost The weekly cost C of producing x units in a manu- facturing process is given by C共x兲 ⫽ 60x ⫹ 750. Year y1 y2 y3 The number of units x produced in t hours is x共t兲 ⫽ 50t. 1995 146 330 457 (a) Find and interpret C共x共t兲兲. 1996 152 344 483 (b) Find the number of units produced in 4 hours. 1997 162 361 503 (c) Use a graphing utility to graph the cost as a function of 1998 176 385 520 time. Use the trace feature to estimate (to two-decimal- 1999 185 414 550 place accuracy) the time that must elapse until the cost 2000 193 451 592 increases to $15,000. 2001 202 497 655 80. Air Traffic Control An air traffic controller spots two 2002 214 550 718 planes at the same altitude flying toward each other. Their 2003 231 601 766 flight paths form a right angle at point P. One plane is 150 miles from point P and is moving at 450 miles per hour. 2004 246 647 824 The other plane is 200 miles from point P and is moving at 2005 262 691 891 450 miles per hour. Write the distance s between the planes as a function of time t. The models for this data are y1 ⴝ 11.4t 1 83, y y2 ⴝ 2.31t 2 ⴚ 8.4t 1 310, and y3 ⴝ 3.03t 2 ⴚ 16.8t 1 467, where t represents the year, with t ⴝ 5 corresponding to Distance (in miles) 1995. 200 75. Use the models and the table feature of a graphing utility to create a table showing the values of y1, y2, and y3 for each s 100 year from 1995 to 2005. Compare these values with the original data. Are the models a good fit? Explain. 76. Use a graphing utility to graph y1, y2, y3, and x P 100 200 yT ⫽ y1 ⫹ y2 ⫹ y3 in the same viewing window. What does the function yT represent? Explain. Distance (in miles) 77. Ripples A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outermost ripple is given by r 共t兲 ⫽ 0.6t, where t is the time (in seconds) after the pebble strikes the water. The area of the circle is given by A共r兲 ⫽ ␲ r 2. Find and interpret 共A ⬚ r兲共t兲. 333371_0106.qxd 12/27/06 10:34 AM Page 146 146 Chapter 1 Functions and Their Graphs 81. Bacteria The number of bacteria in a refrigerated food 86. If you are given two functions f 共x兲 and g共x兲, you can product is given by N共T兲 ⫽ 10T 2 ⫺ 20T ⫹ 600, for calculate 共 f ⬚ g兲共x兲 if and only if the range of g is a subset 1 ≤ T ≤ 20, where T is the temperature of the food in of the domain of f. degrees Celsius. When the food is removed from the refrig- erator, the temperature of the food is given by Exploration In Exercises 87 and 88, three siblings are of T共t兲 ⫽ 2t ⫹ 1, where t is the time in hours. three different ages. The oldest is twice the age of the mid- (a) Find the composite function N共T共t兲兲 or 共N ⬚ T兲共t兲 and dle sibling, and the middle sibling is six years older than interpret its meaning in the context of the situation. one-half the age of the youngest. (b) Find 共N ⬚ T兲共6兲 and interpret its meaning. 87. (a) Write a composite function that gives the oldest sib- (c) Find the time when the bacteria count reaches 800. ling’s age in terms of the youngest. Explain how you 82. Pollution The spread of a contaminant is increasing in a arrived at your answer. circular pattern on the surface of a lake. The radius of the (b) If the oldest sibling is 16 years old, find the ages of the contaminant can be modeled by r共t兲 ⫽ 5.25冪t, where r is other two siblings. the radius in meters and t is time in hours since contamina- 88. (a) Write a composite function that gives the youngest tion. sibling’s age in terms of the oldest. Explain how you (a) Find a function that gives the area A of the circular leak arrived at your answer. in terms of the time t since the spread began. (b) If the youngest sibling is two years old, find the ages of (b) Find the size of the contaminated area after 36 hours. the other two siblings. (c) Find when the size of the contaminated area is 6250 89. Proof Prove that the product of two odd functions is an square meters. even function, and that the product of two even functions is 83. Salary You are a sales representative for an automobile an even function. manufacturer. You are paid an annual salary plus a bonus of 90. Conjecture Use examples to hypothesize whether the 3% of your sales over $500,000. Consider the two func- product of an odd function and an even function is even or tions odd. Then prove your hypothesis. f 共x兲 ⫽ x ⫺ 500,000 and g(x) ⫽ 0.03x. 91. Proof Given a function f, prove that g共x兲 is even and If x is greater than $500,000, which of the following repre- h共x兲 is odd, where g共x兲 ⫽ 12 关 f 共x兲 ⫹ f 共⫺x兲兴 and sents your bonus? Explain. h共x兲 ⫽ 12 关 f 共x兲 ⫺ f 共⫺x兲兴. (a) f 共g共x兲兲 (b) g共 f 共x兲兲 92. (a) Use the result of Exercise 91 to prove that any function 84. Consumer Awareness The suggested retail price of a new can be written as a sum of even and odd functions. car is p dollars. The dealership advertised a factory rebate (Hint: Add the two equations in Exercise 91.) of $1200 and an 8% discount. (b) Use the result of part (a) to write each function as a (a) Write a function R in terms of p giving the cost of the sum of even and odd functions. car after receiving the rebate from the factory. 1 f 共x兲 ⫽ x 2 ⫺ 2x ⫹ 1, g 共x兲 ⫽ (b) Write a function S in terms of p giving the cost of the x⫹1 car after receiving the dealership discount. (c) Form the composite functions 共R ⬚ S 兲共 p兲 and 共S ⬚ R兲共 p兲 Skills Review and interpret each. In Exercises 93– 96, find three points that lie on the graph of (d) Find 共R ⬚ S兲共18,400兲 and 共S ⬚ R兲共18,400兲. Which yields the equation. (There are many correct answers.) the lower cost for the car? Explain. 93. y ⫽ ⫺x2 ⫹ x ⫺ 5 94. y ⫽ 15 x3 ⫺ 4x2 ⫹ 1 Synthesis x 95. x2 ⫹ y2 ⫽ 24 96. y ⫽ x2 ⫺ 5 True or False? In Exercises 85 and 86, determine whether the statement is true or false. Justify your answer. In Exercises 97–100, find an equation of the line that passes 85. If f 共x兲 ⫽ x ⫹ 1 and g共x兲 ⫽ 6x, then through the two points. 共 f ⬚ g兲共x兲 ⫽ 共g ⬚ f 兲共x兲. 97. 共⫺4, ⫺2兲, 共⫺3, 8兲 98. 共1, 5兲, 共⫺8, 2兲 99. 共 3 2, ⫺1兲, 共 ⫺ 13, 4兲 100. 共0, 1.1兲, 共⫺4, 3.1兲