3 CHAPTER Functions Working Together In this chapter we’ll look at ways of combining functions in order to construct new functions. 3.1 COMBINING OUTPUTS: ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF FUNCTIONS The sum, difference, product, and quotient of functions are the new functions defined respectively by the addition, subtraction, multiplication, and division of the outputs or values of the original functions. Addition and Subtraction of Functions ◆ EXAMPLE 3.1 Suppose a company produces widgets. 1 The revenue (money) the company takes in by selling widgets is a function of x, where x is the number of widgets produced. We call this function R(x). We call C(x) the cost of producing x widgets. Producing and selling x widgets results in a profit, P(x),where profit is revenue minus cost; P(x)= R(x) − C(x). The height of the graph of the profit function is obtained by subtracting the height of the cost function from the height of the revenue function. Where P(x)is negative the company loses money. The x-intercept of the P(x)graph corresponds to the break-even point where revenue exactly equals costs, so there is zero profit. (See Figure 3.1 on the following page.) 1 A widget is an imaginary generic product frequently used by economists when discussing hypothetical companies. 101 102 CHAPTER 3 Functions Working Together x C(x) 100 R(x) –F 100 x dollars P(x)=R(x)-C(x) dollars F Figure 3.1 ◆ More generally, if h is the sum of functions f and g, h(x) = f(x)+g(x), then the output of h corresponding to an input of x 1 is the sum of f(x 1 )and g(x 1 ). In terms of the graphs, the height of h at x 1 is the sum of the heights of the graphs of f and g at x 1 .An analogous statement can be made for subtraction. The domain of h is the set of all x common to the domains of both f and g. ◆ EXAMPLE 3.2 Let f(x)=x and g(x) = 1 x . We are familiar with the graphs of f and g. We can obtain a rough sketch of f(x)+g(x) = x + 1 x from the graphs of f and g by adding together the values of the functions as shown in the figure below. x g x f+g x f + => sum the heights Figure 3.2 ◆ EXERCISE 3.1 The following questions refer to Example 3.2, where f(x)+g(x) = x + 1 x . (a) What is the domain of f + g? (b) For |x| close to zero, which term of the sum dominates (controls the behavior of) the sum? (c) For |x| large, which term of the sum is dominant? EXERCISE 3.2 When a company produces widgets they have fixed costs (costs they incur regardless of whether or not they produce a single widget, such as renting some space for widget production) and they have variable costs (costs that vary with the number of widgets they produce, such as the cost of materials and labor). Figure 3.3 shows graphs of the fixed cost function, FC,and the variable cost function, VC,for widgets. Total cost = fixed costs + variable costs Graph the total cost function, TC,where TC=FC +VC. 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions 103 x F dollars FC VC Figure 3.3 EXERCISE 3.3 A patient is receiving medicine intravenously. Below are two graphs. One is the graph of R I (t), the rate at which the medication enters the bloodstream. The other is a graph of R O (t), the rate at which the medication is metabolized and leaves the bloodstream. 2 (a) Let R(t) be the rate of change of medication in the bloodstream. Express R(t) in terms of R I (t) and R O (t). (b) Graph R(t). (c) At approximately what value of t is R(t) minimum? 1 2 3 4 5 1 2 3 4 5 t t R I R O 0 0 Figure 3.4 Multiplication and Division of Functions Let’s suppose a consultant wants to construct a function to model the amount of money he spends on gasoline for his automobile. The price of gasoline varies with time. Let’s denote it by p(t), where p(t) is measured in dollars per gallon. The amount of gasoline he uses for his commute also varies with time; we’ll denote it by g(t). Then the amount of money he spends is given by p(t) $ gal · g(t) gal. This is the product of the functions p and g. A demographer is interested in the changing economic profile of a certain town. If the population of the town at time t is given by P(t)and the total aggregate income of the town at time t is given by I(t),then the per capita income is given by 2 R I for “rate in”; R O for “rate out.” 104 CHAPTER 3 Functions Working Together I(t) P(t) . This is the quotient of the functions I and P . If h(x) = f(x)· g(x), then the output of h is the product of the outputs of f and g.If j(x)= f(x) g(x) , then the output of j is the quotient of the outputs of f and g. The product is defined for any x in the domains of both f and g; the quotient is defined for any x in the domains of both f and g provided g(x) is not equal to 0. EXERCISE 3.4 h(x) = f(x)· g(x) (a) If h(a) > 0, what can be said about the signs of f and g at x = a? (b) If h(a) < 0, what can be said about the signs of f and g at x = a? (c) How are the zeros of h related to the zeros of f and g? EXERCISE 3.5 j(x)= f(x) g(x) (a) If j(a)>0, what can be said about the signs of f and g at x = a? (b) How are the zeros of j related to the zeros of f and g? ◆ EXAMPLE 3.3 The number of widgets people will buy depends on the price of a widget. Economists call the number of widgets people will buy the demand for widgets. Thus, demand for a widget is a function of price. Let’s suppose that the number of widgets demanded is given by D(p), where p is the price of a widget. If a company has a monopoly on widgets, then it can fix the price of a widget to be whatever it likes. The revenue, R, that this company takes in is given by (price of a widget) · (number of widgets sold), so R(p) = p · D(p). Below is the graph of the demand function, where quantity demanded is a function of price. 3 price quantity demanded p 1 Figure 3.5 (a) What prices will yield no revenue? Why? (b) Sketch a rough graph of R(p). 3 Economists would reverse the axes. 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions 105 SOLUTION (a) Prices of $0 and $p 1 will yield no revenue. (In fact, any price above $p 1 will yield no revenue.) If widgets are free there is no revenue, and likewise if the price of a widget is $p 1 or more then nobody will buy widgets, so the revenue is also zero. (b) The graph of R(p) is given below. R(p) p R(p) p 1 Figure 3.6 ◆ EXERCISE 3.6 Let C(q) be the total cost of making q widgets. Assume that C(0) is positive. How would you compute the average total cost of making q widgets? When graphing the average total cost function, what units might be on the coordinate axes? Why is it that average total cost curves never intersect either of the two coordinate axes? EXERCISE 3.7 Let f(x)=x and g(x) = 1/x. Sketch the graph of h(x) = f(x)·g(x). Where is h(x) undefined? How can you indicate this on your graph? How does your graphing calculator deal with the point at which h is undefined? The answer to Exercise 3.7 is supplied at the end of the chapter. PROBLEMS FOR SECTION 3.1 1. Let f(x)=x 2 and g(x) = 1/x. Use your knowledge of the graphs of f and g to sketch the graph of h(x) = f(x)·g(x). Where is h(x) undefined? How can you indicate this on your graph? How does your graphing calculator deal with the point at which h is undefined? 2. Let f(x)=|x|and g(x) = 1/x. Use your knowledge of the graphs of f and g to sketch the graph of h(x) = f(x)·g(x). Where is h(x) undefined? Note: You must deal with the cases x>0and x<0separately. This is standard protocol for handling absolute values. 3. Let f(x)=|x|and g(x) = x. Use your knowledge of the graphs of f and g to sketch the graph of h(x) = f(x)+g(x). 4. Let B(t) denote the birth rate of Siamese fighting fish as a function of time and D(t) denote the death rate. Then the total rate of change of the population of Siamese fighting fish, R(t), is given by subtracting the death rate from the birth rate; thus, 106 CHAPTER 3 Functions Working Together R(t) = B(t) − D(t). Graphs of B(t) and D(t) are shown below. Sketch a graph of R(t). t (months) D(t) fish /month B(t) 5. Let F(t) be the number of trout in a given lake as a function of time and suppose that K(t) is the fraction of these fish in the lake at time t that are “keepers” if caught (“keepers” meaning that they are above a certain minimum length—smaller ones are thrown back). Then the total number of keepers in the lake at any time is given by the product of F(t)and K(t). Below are graphs of F(t)and K(t). Sketch a graph of N(t), the total number of keepers as a function of time. 2000 3000 t F(t) t K(t) 40% 6. A town draws its water from the town reservoir. The town’s water needs vary throughout the day; the rate of water leaving the reservoir (in gallons per hour) is shown on the graph below. Also recorded is the rate at which water is flowing into the reservoir from a nearby stream. rate of water running out rate of water running in time 7:00 8:00 9:00 10:00 11:00 noon 1:00 2:00 3:00 4:00 5:00 6:00 7:00 500 1000 1500 2000 2500 3000 gallons/hr (a) At 6:00 a.m., at what rate is the water being used by the town? At what rate is water flowing in from the stream? Is the water level in the reservoir increasing or decreasing at 6:00 a.m.? At what rate? 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions 107 (b) At approximately what time(s) is the rate of flow of water into the reservoir equal to the rate of flow out of the reservoir? (c) During what hours (between 6:00 a.m. and 7:00 p.m.) is the water level in the reservoir increasing? (d) At approximately what time is the water level in the reservoir increasing most rapidly? How can you get this information from the graph? 7. Let’s return to the city reservoir in Problem 6. We’ll denote the rate at which water is flowing into the reservoir by R I (t) and the rate that it is flowing out by R O (t). Then the total rate of change of water in the reservoir is given by Total rate of change = Rate in − Rate out or R(t) = R I (t) − R O (t). Graph R(t). 8. Let f(x)=x(x + 1), g(x) = x 3 + 2x 2 + x. (a) Simplify the following. i. f(x)+g(x) ii. f(x) g(x) iii. g(x) f(x) iv. [f(x)] 2 g(x) (b) Solve xf (x) = g(x). 9. The Cambridge Widget Company is producing widgets. The fixed costs for the com- pany (costs for rent, equipment, etc.) are $20,000. This means that before any widgets are produced, the company must spend $20,000. Suppose that each widget produced costs the company an additional $10. Let x equal the number of widgets the company produced. (a) Write a total cost function, C(x),that gives the cost of producing x widgets. (Check that your function works, e.g., check that C(1) = 20, 010 and C(2) = 20, 020.) Graph C(x). (b) At what rate is the total cost increasing with the production of each widget? In other words, find C/x. (c) Suppose the company sells widgets for $50 each. Write a revenue function, R(x), that tells us the revenue received from selling x widgets. Graph R(x). (d) Profit = total revenue − total cost, so the profit function, P(x), which tells us the profit the company gets by producing and selling x widgets, can be found by computing R(x) − C(x). Write the profit function and graph it. (e) Find P(400) and P(700); interpret your answers. Find P(401) and P(402).By how much does the profit increase for each additional widget sold? Is P /x constant for all values of x? (f) How many widgets must the company sell in order to break even? (Breaking even means that the profit is 0; the total cost is equal to the total revenue.) (g) Suppose the Cambridge Widget Company has the equipment to produce at maxi- mum 1200 widgets. Then the domain of the profit function is all integers x where 108 CHAPTER 3 Functions Working Together 0 ≤ x ≤ 1200. What is the range? How many widgets should be produced and sold in order to maximize the company’s profits? 10. A photocopying shop has a fixed cost of operation of $4000 per month. In addition, it costs them $0.01 per page they copy. They charge customers $0.07 per page. (a) Write a formula for R(x), the shop’s monthly revenue from making x copies. (b) Write a formula for C(x), the shop’s monthly costs from making x copies. (c) Write a formula for P(x), the shop’s monthly profit (or loss if negative) from making x copies. Profit is computed by subtracting total costs from the total revenue. (d) How many copies must they make per month in order to break even? Breaking even means that the profit is zero; the total costs and total revenue are equal. (e) Sketch C(x), R(x), and P(x) on the same set of axes and label the break-even point. (f) Find a formula for A(x), the shop’s average cost per copy. (g) Make a table of A(x) for x = 0, 1, 10, 100, 1000, 10000. (h) Sketch a graph of A(x). 3.2 COMPOSITION OF FUNCTIONS Whereas the addition, subtraction, multiplication, and division of functions is simply the addition, subtraction, multiplication, and division of the outputs of these functions, another way of having functions work together is to have the output of one function used as the input of the next function. Suppose you are blowing up balloons for a celebration. The surface area S of the balloon is a function of a, the amount of air inside of the balloon. Let’s say S = f(a).The amount of air inside the balloon is a function of time. Let’s say a = g(t). Then S = f(a)=f(g(t)). Wecan say that S is equal to the composition of f and g. Composition of functions is analogous to setting up functions as workers (or machines) on an assembly line. If the output of g is handed over to f as input, we write f(g(t)), where the notation indicates that f acts on the output of g. This can be represented diagrammatically by t g −→ g(t) f −→ f(g(t)). The expression f(g(t))says “apply f to g(t)”; that is, use g(t) as the input of f . This is called the composition of f and g} and is also sometimes written as f ◦ g. The expression (f ◦ g)(t) means f(g(t)):Start with t ; apply machine g and then apply machine f to the result. 3.2 Composition of Functions 109 tg(t) f f(g(t))=(f◦g)(t) g Caution: Do not let the notation mislead you into thinking that f should be applied before g; (f ◦ g)(t) means apply g, then apply f to the result. The domain of f ◦ g is the set of all t in the domain of g such that g(t) is in the domain of f . Notice that the order in which machines are put on an assembly line is generally critical to the outcome of the process. Suppose machine W pours a liter of water in a specified place and machine L places a lid on a bottle. We send an open empty bottle down the assembly line. Putting machine W first on the assembly line, followed by machine L, results in the production of bottled water, while reversing the order results in the production of sealed, washed, empty bottles. Open empty bottle W −→ L −→ bottled water corresponding to L(W (bottle)) Open empty bottle L −→ W −→ w ashed bottle corresponding to W (L(bottle)) From this example we see that generally f(g(t))=g(f (t)). When unraveling the composition of functions, always start from the innermost paren- theses and work your way outward. This will assure the correct order on the assembly line of functions. ◆ EXAMPLE 3.4 Let f(x)=x 2 , g(x) = 2x + 3. Find the following. i. (f ◦ g)(x), i.e., f(g(x)) ii. (g ◦ f )(x), i.e., g(f (x)) SOLUTION i. (f ◦ g)(x) = f(g(x)).Toevaluate, replace g(x) by its output value, which is 2x + 3. Next, treat (2x + 3) as the input of the function f ; f squares the input. f(g(x))=f(2x+3)=(2x +3) 2 =4x 2 +12x + 9 ii. (g ◦ f )(x) = g(f (x)) = g(x 2 ) = 2x 2 + 3 ◆ Notice that in Example 3.4 for almost all values of x, f(g(x))=g(f (x)). 4 The order in which the functions are composed determines the result. In this case, doubling the input, adding 3 to it, and then squaring the sum is different from squaring the input, doubling the result, and then adding 3. When we write mathematics, we indicate the order of operations in an expression through a combination of parentheses and conventions for orders of operations. 5 ◆ EXAMPLE 3.5 Let f(x)=x 2 ,g(x) = 2x + 3, as in Example 3.4. Find i. f (g(g(x))) ii. g  1 f(x)  4 There are only two values of x for which 2x 2 + 3 is the same as 4x 2 + 12x + 9. See if you can find them. If you need a refresher, refer to Appendix A: Algebra, under Solving Quadratic Equations. 5 For a review of conventions for order of operations, please refer to Appendix A: Algebra. 110 CHAPTER 3 Functions Working Together SOLUTION i. f (g(g(x))) = f(g(2x+3)) = f(2(2x+3)+3) =f(4x+6+3) =f(4x+9) =(4x+9) 2 =16x 2 + 72x + 81 ii. g  1 f(x)  = g  1 x 2  = 2 x 2 + 3 ◆ ◆ EXAMPLE 3.6 Let g(x) = 2x + 3, h(x) = x−3 2 . Find i. h(g(x)) ii. g(h(x)) SOLUTION i. h(g(x)) = h(2x + 3) = (2x+3)−3 2 = 2x 2 = x ii. g(h(x)) = g  x−3 2  = 2  x−3 2  + 3 = x − 3 + 3 = x ◆ Observation In this example diagrammatically we have x g −→ g(x) h −→ x and x h −→ h(x) g −→ x . Thus, whether g is followed by h or h is followed by g, the result is not only the same, but it is the original input. The functions h and g undo one another. If h(g(x)) = x and g(h(x)) = x, then h and g are called inverse functions.(Wefirst introduced the topic of inverse functions in Section 1.3 and will discuss it in detail in Chapter 12.) Notice that in order to perform the composition of functions you need to be comfortable evaluating a function even when the input is rather messy. You must distinguish in your mind the difference between the functional rule itself and the input of the function. The following exercise may be helpful. EXERCISE 3.8 Let f be the function given by f(x)= x x−1 + 2x. Find the following. i. f(3) ii. f(y +1) iii. f(1/x) iv. f(x +h) v. f(2h) h To do this exercise, it is important to keep in mind that whatever is enclosed in the parentheses of f is the input of f . What does the function do with its input? f divides the input by a number that is one less than the input and then adds twice the input to that to the quotient. For a silly but fail-proof way to find f (mess), run through the following questions: What is f(2)?f(3)?f(π)?f(mess)? This serves to get the functional rule firmly established in your mind. Solutions to Exercise 3.8 are given at the end of the chapter. . the output of h corresponding to an input of x 1 is the sum of f(x 1 )and g(x 1 ). In terms of the graphs, the height of h at x 1 is the sum of the heights of the graphs of f and g at x 1 .An analogous. f(a)=f(g(t)). Wecan say that S is equal to the composition of f and g. Composition of functions is analogous to setting up functions as workers (or machines) on an assembly line. If the output of g is handed. x<0separately. This is standard protocol for handling absolute values. 3. Let f(x)=|x |and g(x) = x. Use your knowledge of the graphs of f and g to sketch the graph of h(x) = f(x)+g(x). 4. Let