3.2 Composition of Functions 111 EXERCISE 3.9 Let f(x)=x 2 and g(x) = x + 1. (a) Find h(x) = f(g(x))and sketch the graph of h. (Use any means at your disposal.) (b) Find j(x)= g(f (x)) and sketch the graph of j. (c) Now look at parts (a) and (b) as variations on the theme f(x)=x 2 . i. In part (a) the input of f is increased by 1. What is the effect on the graph? ii. In part (b) the output of f is increased by 1. What is the effect on the graph? EXERCISE 3.10 Let f(x)=1/x and g(x) = x + 2. (a) Find h(x) = f(g(x))and sketch the graph of h. (b) Find j(x)= g(f (x)) and sketch the graph of j. (c) Now look at parts (a) and (b) as variations on the theme f(x)=1/x. i. In part (a) the input of f is increased by 2. What is the effect on the graph? ii. In part (b) the output of f is increased by 2. What is the effect on the graph? ◆ EXAMPLE 3.7 An oil spill on a large lake. Let’s return to our model of an oil spill where oil spreads out into a thin expanding disk. In Chapter 2 we looked at how the area of the disk of oil varies with the radius. Of more practical importance is how the area of the spill varies with time. Is the leak steady, worsening, or slowing down? Suppose the cannister from which the oil is leaking is opaque so we can’t take our measurement directly from there. Instead, we measure the radius of the spill (in yards) at various times. The idea is to express the radius r as a function of time, r = g(t). Then, since we can express the area of the spill as a function of the radius, A = f(r)=πr 2 , wewill be able to replace r by g(t), obtaining A = f(g(t)). The composition f(g(t))allows us to express the area as a function of time. Suppose we measure the radius of the spill at one-hour intervals. The information is recorded below. 6 Time (hr) 012345678910 Radius 0.00 1.00 1.41 1.73 2.00 2.23 2.44 2.65 2.83 3.00 3.16 A plot of the results of our measurements is given in Figure 3.7. The graph of r versus t should be modeled by a continuous function because r is continuously increasing. 6 Note that if measurements had been taken just for t = 4, 5, 6, 7, 8, and 9 and if the radius of the spill was recorded to only one decimal place, it would appear that each hour the radius increased by two tenths of a yard, and that the radius was a linear function of time. You can see this by looking at the graph in Figure 3.7 on the following page and seeing that the points above t = 4, 5, 6, 7, 8, and 9 almost appear to lie on a single line. In fact, it is not linear, but locally (on small enough intervals) it is approximately linear. We say that it is locally linear. The notion of local linearity is central to differential calculus and will be discussed in more detail in Chapter 4. 112 CHAPTER 3 Functions Working Together 1 2 3 4 5 6 7 8 9 10 t (hours) 1 2 3 r (radius) Figure 3.7 t r r(t) 12345678910 1 2 3 Figure 3.8 Answer the following questions. i. How does the radius vary with time? Give a possible model for r as a function of t.In other words, find a function g(t) that would have a graph similar to that in Figure 3.8. ii. Using your answer to part (i), express area as a function of time. iii. How is the area changing with respect to time? What does this tell us about the oil leak? SOLUTION i. r = g(t) = √ t looks like an excellent model for t for 0 ≤ t ≤ 10. ii. Area of a circle = f(r)=πr 2 . Using part (i), we have r = g(t) = √ t ,so A=f(r)=f(g(t))=f( √ t) =π( √ t) 2 =πt. t 0123 45678910 r 0.00 1.00 1.41 1.73 2.00 2.23 2.44 2.65 2.83 3.00 3.16 A 0.00 3.14 6.28 9.42 12.57 15.71 18.85 21.99 25.13 28.27 31.42 t → r →A gf (The bottom row in this table is πt rounded to two decimal places.) 3.2 Composition of Functions 113 iii. Once we know A as a function of r and know r as a function of t we can find A as a function of t using composition of functions. A(t) = πt, so each hour the area of the spill increases by π square yards. This indicates that the leak is steady. It shows no signs of diminishing or speeding up; oil is leaking out at a steady rate. The rate at which the area is increasing is π square yards/hour; this is approximately 3.14 sq yd/hr. Mental Picture: Assembly Line Model Greta's job Fred's job fg g(t) f(g(t)) t In this assembly line, g stands for Greta’s job. At each time t, Greta determines the radius of the spill, g(t). Greta hands her output, the radius, over to Fred, who uses it to compute the area of the spill. Fred’s job, f , is to take what he is given, square it, and multiply by π. t → g(t) →f(g(t))=f( √ t) =π( √ t) 2 =πt ◆ PROBLEMS FOR SECTION 3.2 1. To find f(g(x)),apply g to x and then use the output of g as the input of f . Work from the inside out. Let f(x)=x 2 ,g(x) =1/x, and h(x) = 3x +1. Worked example: f (g(h(x))) = f(g(3x+1)) = f  1 3x +1  =  1 3x +1  2 Find the following. (a) f(g(x)) (b) g(f(x)) (c) xh(f (x)) (d) f (h(g(x))) (e) g(g(w)) (f) h(h(t)) (g) g(f(1/x)) (h) g(2h(x − 1)) (i) Show that g(g(x)) =[g(x)] 2 (j) Show that [h(x)] 2 = h(x 2 ). For Problems 2 through 10, f and g are functions with domain [−3, 4]. Their graphs are provided below. Use the graphs to approximate the following. –3 –2 –11324 4 3 2 1 – 4 –3 –2 –1 f x –3 –2 –11324 4 3 2 1 – 4 –3 –2 –1 g x 114 CHAPTER 3 Functions Working Together 2. (a) f(g(1)) (b) f(g(0)) 3. (a) g(f(1)) (b) f(g(0)) 4. (a) f(f(2)) (b) f(f(1)) 5. (a) g(g(0)) (b) f(f(0)) 6. Find x such that 2f(x)=0. 7. Find all x such that f(g(x))=0. 8. Find all x such that g(f (x)) =0. 9. (a) Find all x such that f(x)=3. (b) Find all x such that f(g(x))=3. 10. Solve for x: f(0)x + f(1)=g(0)x + g(1). For Problems 11 through 13, find f (g(h(x))) and g(h(f (x))). 11. f(x)= 1 x , g(x) = √ x, h(x) = x −3 12. f(x)=3x 2 +x,g(x) =x + 1, h(x) = 2 3x 13. f(x)=x +2, g(x) = x 2 , h(x) = x 2−x For Problems 14 and 15 let f(x)=x −3and g(x) =x 2 −6x. 14. Evaluate and simplify each of the following expressions. (a) f(x)+g(x) (b) f(x)−g(x) (c) f (x)g(x) (d) f(g(x)) (e) g(f(x)) (f) f(x) g(x) 15. Find the x- and y-intercepts of the following. (a) f(x) (b) g(x) (c) f (x)g(x) (d) f(x) g(x) 16. How do the x- and y-intercepts of f(x) and g(x) affect the intercepts of f (x)g(x)? State this as a general rule. 17. For what values of x (if any) is f(x) g(x) undefined? 18. How do the x- and y-intercepts of f(x)and g(x) affect the intercepts of f(x) g(x) and the places where f(x) g(x) is undefined? 19. Suppose that the functions f , g, and h are defined for all integers. At the top of the following page is a table of some of the values of these functions. 3.2 Composition of Functions 115 x −2 −1 012 34 f(x) 0 2 134−25 g(x) 2 3 413−10 h(x) 34−328 12 Evaluate the following expressions. If not enough information is available for you to do so, indicate that. (a) f(−1)·g(−1) (b) f(g(−1)) (c) g(f(−1)) (d) h(g(f (2))) (e) f(0)+2 g(0) (f) 5h(3) + f(f(1)) (g) f(f(f(0))) 20. The graphs of f(x)and g(x) are given below. (a) Approximate all the zeros of the function h(x) = f(g(x)). (b) Approximate all the zeros of the function j(x)=g(f (x)). f – 4 – 4 –3 –3 –2 –2 –1 –1 1 1 3 3 2 2 4 4 x g – 4 – 4 –3 –3 –2 –2 –1 –1 1 1 3 3 2 2 4 4 x 21. You put $300 in a bank account at 4% annual interest compounded annually and you plan to leave it there without making any additional deposits or withdrawals. With each passing year, the amount of money in the account is 104% of what it was the previous year. (a) Write a formula for the function f that takes as input the balance in the account at some particular time and gives as output the balance one year later. Write this formula as one term, not the sum of two terms. (b) Two years after the initial deposit is made, the balance in the account is f(f(300)) and three years after, it is f(f(f(300))). Explain. (c) What quantity is given by f(f(f(f(300))))? (d) Challenge: Write a formula for the function g that takes as input n, the number of years the deposit of M dollars has been in the bank, and gives as output the balance in the account. 116 CHAPTER 3 Functions Working Together 22. f(x)= 1 2−x and g(x) =x 2 +1. Find the following. Simplify your answers. If simpli- fying is difficult, consult Appendix A: Algebra. (a) 2f(x +1) (b) f(2x−2) (c) g( √ x +1) (d) f(g(x)) (e) g(f(x)) (f) f(f(x)) (g) g  1 f(x)  (h) g(x) f(x) 23. Most of the time, when a store provides coupons offering $5 off any item in the store they include the clause “except for sale items.” Suppose that clause were omitted and you found an item you wanted on a “30% off” rack. There would be some ambiguity; should the $5 be taken off the reduced price, or off the price before the 30% discount? Let C be the function that models the effect of the coupon, S be the function that models the effect of the sale, and x be the original price of the item. (a) Which situation corresponds to C(S(x))? (b) Which situation corresponds to S(C(x))? (c) Which order of composition of the functions is in the buyer’s favor? 24. Two brothers, Max and Eli, are experimenting with their walkie-talkies. (A walkie- talkie is a combined radio transmitter and receiver light enough to allow the user to walk and talk at the same time.) The quality of the transmission, Q, is a function of the distance between the two walkie-talkies. We will model it as being inversely proportional to this distance. At time t = 0 Max is 100 feet north of Eli. Max walks north at a speed of 300 feet per minute while Eli walks east at a speed of 250 feet per minute. All the time they are talking on their walkie-talkies. (a) Write a function f such that Q = f(d), where d is the distance between the brothers. Your function will involve an unknown constant. (b) Write a function g that gives the distance between the brothers at time t. (c) Find f(g(t)). What does this composite function take as input and what does it give as output? 25. If h(x) = f(g(x)),then x is in the domain of h if and only if x is in the domain of g and g(x) is in the domain of f . In other words, x must be a valid input for g and g(x) must be a valid input for f . (a) If h(x) = f(g(x)),where g(x) = √ x and f(x)=x 2 ,what is the largest possible domain of h? For all x in its domain, h(x) = x. Why is the domain not (−∞, ∞)? (b) If h(x) =f(g(x)),where g(x) = 1 x−1 and f(x)= 1 x+3 , what is the largest possible domain of h? (There are two numbers that must be excluded from the domain.) 26. Let f(x)= 2x x+3 and g(x) = 1 x+1 . (a) Find f(g(2)). (b) Find f(g(x))and simplify your answer. Be sure that your answer is in agreement with the concrete case from part (a). 27. Let f(x)= x x+3 and g(x) = 3x 1−x . (a) Find f(g(2)) and g(f (2)). (b) Find f(g(x))and g(f (x)). (c) What does part (b) suggest about the relationship between f and g? 3.2 Composition of Functions 117 28. Below are graphs of f and g. –3 –2 –11324 3 2 1 –2 –1 f x –3 –2 –11324 3 2 1 –2 –1 g x (a) Approximate f(g(2)). (b) Approximate g(f(2)). (c) For what values of x is f(g(x))=0? 29. If the function m(t) = 1 t+2 and h(t) =t − 2, then is it ever true that m(h(t)) =h(m(t))? 30. The functions R(x), K(x), D(x), and L(x) are defined as follows: R(x) = 1 x 2 , K(x) =|x|, D(x) =x +3, L(x) =−5x. Evaluate the following expressions. (Be sure to give simplified expressions whenever possible.) (a) R(K(L(x))) (b) R(L(R(x))) (c) R(K(x)) (d) R(D(R(x))) In Problems 31 through 33, let f(x)=|x|,g(x) = √ x, and h(x) = x − 2. Find the domain for each of the following. 31. (a) j(x)=g(h(x)) (b) k(x) =h(g(x)) 32. (a) l(x) =g(f(x)) (b) m(x) = g(h(f (x))) 33. (a) p(x) = h(g(h(x))) (b) q(x) =f (h(g(x))) 34. Let f(x)=x 2 +9, g(x) = √ x, and h(x) = g(f (x)). Find the average rate of change of h over the following intervals. (a) [−4, 4] (b) [0, 4] (c) [4, 4 + k] In Problems 35 through 38, find h(x) = f(g(x)) and j(x)= g(f(x)). What are the domains of h and j ? 35. f(x)=x 2 +9and g(x) = 1 √ x 36. f(x)= 2 x+2 and g(x) =x −2 118 CHAPTER 3 Functions Working Together 37. f(x)=x 2 and g(x) =−2x+3 38. f(x)= x x−3 and g(x) = 2 x In Problems 39 through 43, find (f + g)(x), (fg)(x), and  f g  (x), and find their domains. 39. f(x)=ax + b and g(x) = cx + d 40. f(x)=3x +2and g(x) = 5x − 1 41. f(x)=2x +3and g(x) =x 2 −1 42. f(x)= 3 x+1 and g(x) = 2x x−5 43. f(x)= √ x and g(x) = √ x −3 In Problems 44 through 49, let f(x)= 1 x + x and g(x) = 2x x 2 +1 . Evaluate the following expressions. 44. (a) g(f(2)) (b) f(g(2)) 45. (a) g(f( 1 3 )) (b) f(g( 1 3 )) 46. (a) g(f(1)) (b) f(g(1)) 47. (a) f(f(2)) (b) g(g(−1)) 48. (a) f(g(x)) (b) g(f(x)) 49. (a) (f ◦f )(x) (b) (f ◦f ◦f )(x) 50. (a) Suppose f and g are both even functions. What can be said about (f + g)(x)? (fg)(x)? (b) Suppose f and g are both odd functions. What can be said about (f + g)(x)? (fg)(x)? (c) Suppose f is an even function and g is an odd function. What can be said about (f + g)(x)? (fg)(x)? Algebraic calisthenics: Let f(x)=2x 2 ,g(x) =x + 1, and h(x) = 1 x . In Problems 51 through 58, if what is written is an expression, simplify it. If it is an equation, solve it. 51. h(f (x)) + h(g(x)) 52. g(x)h(f (x)) + f (x)h(g(x)) 53. g(x)h(f (x)) = 1 54. 2h(f (x)g(x)) + h(3g(x)) 3.3 Decomposition of Functions 119 55. h(f (x) + 3g(x)) =h(2) 56. f(g(x))=10 57. f(g(f(x)))=8 58. f(g(−5+f(x))) = 8 (There are four solutions.) 3.3 DECOMPOSITION OF FUNCTIONS To decompose a function means to express it as the composition of two or more functions. Using our assembly line analogy, the process of decomposition corresponds to breaking up a task into a sequence of simpler jobs to be done in succession on an assembly line, the output of one operation constituting the input for the next. Just as there may be different ways of setting up an assembly line to accomplish a given task, there are often different ways to decompose a single function. We give some examples below. ◆ EXAMPLE 3.8 If h(x) = √ x 2 + 7x, find f and g such that f(g(x))=h(x). Diagrammatically we have the following situation: h x → g(x) → f(g(x)) g f Think of g as the first worker on the assembly line; the output of g is passed on for f to act upon. There are different ways we can set up the assembly line. One possibility is that the first worker, g, can produce x 2 + 7x and, to finish off the job, f can take a square root. Then g(x) =x 2 +7x and f(x)= √ x. Sometimes this is written g(x) =x 2 +7x and f (u) = √ u, this notation (i.e., the different variable) reminding us that f works on the output of another function. ◆ EXERCISE 3.11 There are many ways of decomposing functions. For instance, suppose h(x) = x 2 + 1 x 2 + 2 and we want to find f and g such that f(g(x))=h(x). Determine which of the following pairs of functions produce the appropriate result. h x → g(x) → f(g(x)) g f 120 CHAPTER 3 Functions Working Together i. g(x) =x 2 , f(x)= x+1 x+2 ii. g(x) =x 2 +1, f(x)= x x+1 iii. g(x) =x 2 +2, f(x)= x−1 x iv. g(x) = x 2 x 2 +2 , f(x)= x+1 x v. g(x) =x 2 +1, f(x)= x x 2 +2 Answers are provided at the end of this chapter. ◆ EXAMPLE 3.9 f(x)=  (x 2 + 2) 2 + 3. Find g, h, and j such that g(h(j (x))) = f(x). SOLUTION f x → j(x) → h( j(x)) → g(h( j(x))) jg h There are many different sets of functions g, h, and j that satisfy g(h(j (x))) = f(x). Forexample, j(x)=x 2 h(x) = (x + 2) 2 + 3 g(x) = √ x or j(x)=(x 2 + 2) 2 h(x) = x + 3 g(x) = √ x or j(x)=x 2 h(x) = (x + 2) 2 g(x) = √ x +3 ◆ Why decompose? Decomposing a function can give us insight into its underlying structure and aid us in dealing with more complex functions. A basic strategy when solving any complex problem is to break the problem down into simpler subproblems and to construct a solution by appropriately building it up from solutions to the subproblems. Determining the skeletal structure of the problem is critical when searching for a strategy for solving a complex problem. ◆ EXAMPLE 3.10 Suppose we want to solve the equation x 4 − 5x 2 + 4 = 0. In general, fourth degree equations are difficult to tackle, but this particular equation can be viewed as a “quadratic in disguise.” By this we mean that the underlying structure is that of a quadratic; that is, if h(x) = x 4 − 5x 2 + 4 then h(x) = f(g(x)),where g(x) =x 2 and f (u) = u 2 − 5u + 4 . To solve the original equation we can let u = x 2 and solve the . (fg)(x), and  f g  (x), and find their domains. 39. f(x)=ax + b and g(x) = cx + d 40. f(x)=3x + 2and g(x) = 5x − 1 41. f(x)=2x + 3and g(x) =x 2 −1 42. f(x)= 3 x+1 and g(x) = 2x x−5 43. f(x)= √ x and. of Functions 111 EXERCISE 3.9 Let f(x)=x 2 and g(x) = x + 1. (a) Find h(x) = f(g(x) )and sketch the graph of h. (Use any means at your disposal.) (b) Find j(x)= g(f (x)) and sketch the graph of. f works on the output of another function. ◆ EXERCISE 3.11 There are many ways of decomposing functions. For instance, suppose h(x) = x 2 + 1 x 2 + 2 and we want to find f and g such that f(g(x))=h(x).