PART I Functions: An Introduction 1 CHAPTER Functions Are Lurking Everywhere 1.1 FUNCTIONS ARE EVERYWHERE Each of us attempts to make sense out of his or her environment; this is a fundamental human endeavor. We think about the variables characterizing our world; we measure these variables and observe how one variable affects another. For instance, a child, in his first years of life, names and categorizes objects, people, and sensations and looks for predictable relationships. As a child discovers that a certain phenomenon precipitates a predictable outcome, the child learns. The child learns that the position of a switch determines whether a lamp is on or off, and that the position of a faucet determines the flow of water into a sink. The novice musician learns that hitting a piano key produces a note, and that which key is hit determines which note is heard. The deterministic relationship between the piano key hit and the resulting note is characteristic of the input-output relationship that is the object of our study in this first chapter. 1 2 CHAPTER 1 Functions Are Lurking Everywhere Mathematical modeling involves constructing mathematical machines that mimic im- portant characteristics of commonly occurring phenomena. Chemists, biologists, environ- mental scientists, economists, physicists, engineers, computer scientists, students, and parents all search for relationships between measurable variables. 1 A chemist might be interested in the relationship between the temperature and the pressure of a gas, an environ- mental scientist in the relationship between use of pesticides and mortality rate of songbirds, a physician in the relationship between the radius of a blood vessel and blood pressure, an economist in the relationship between the quantity of an item purchased and its price, a grant manager in the relationship between funds allocated to a program and results achieved. A thermometer manufacturer must know the relationship between the temperature and the volume of a gram of mercury in order to calibrate a thermometer. The list is endless. As human beings trying to make sense of a complex world, we instinctively try to identify relationships between variables. We will concern ourselves here with relationships that can be structured as input- output relationships with the special characteristic that the input completely determines the output. For example, consider the relationship between the temperature and the volume of a gram of mercury. We can structure this relationship by considering the input variable to be temperature and the output variable to be volume. A specific temperature is the input; the output is the volume of one gram of mercury at that specific temperature. The temperature determines the volume. As another example, consider a hot-drink machine. If your inputs are inserting a dollar bill and pressing the button labeled “hot chocolate,” the output will be a cup of hot cocoa and 55¢ in change. In such a machine the input completely determines the output. The mathematical machine used to model such relations is called a function. Mathematicians define a function as a relationship of inputs and outputs in which each input is associated with exactly one output. Notice that the mathematical use of the word “function” and its use in colloquial English are not identical. In colloquial English we might say, “The number of hours it takes to drive from Boston to New York City is a function of the time one departs Boston.” By this we mean that the trip length depends on the time of departure. But the trip length is not uniquely determined by the departure time; holiday traffic, accidents, and road construction play roles. Therefore, in a mathematical sense the length of the trip is not a function of the departure time. Think about the task of calibrating a bottle, marking it so that it can subsequently be used for measuring. The calibration function takes a volume as input and gives a height as output. For any particular bottle we can say that the height of the liquid in the bottle is a function of the volume of the liquid; that is, height (output) is completely determined by volume (input). We use pictures to illustrate the relationship. Tracking the input variable along the horizontal axis and the output variable along the vertical axis is a mathematical convention for displaying graphs of functions. 1 While physicists hope to uncover physical laws, economists and other social scientists often aim for some working under- standing that can be applied appropriately. 1.1 Functions Are Everywhere 3 Height (output) Volume (input) Figure 1.1 Calibration of conical flask The concept of a function is important and versatile. We can attach many different mental pictures and representations to it; our choice of representation will often depend upon context. Upon first exposure, the notation and representations may be confusing, but as you use the language of functions the nuances will become as natural as nuances in the English language. Aside: In any discipline accurate communication is critical; in mathematics a great deal of care goes into definitions because the field is structured so that the validity of arguments rests on logic, definitions, and a few postulates. Definitions arise because they are needed in order to make precise, unambiguous statements. Sometimes a formal definition can initially seem unnatural to you, usually because the person who made the definition wanted to be sure to include (or exclude) a certain situation that hasn’t yet crossed your mind. To circumvent this problem, in this text we will sometimes begin with an informal definition and formalize it later. Just in case you are not kindly disposed to learning the language of functions, we’ll take a brief foray into English word usage to put mathematical language in perspective. Consider the word “subway,” meaning an underground railway. The word conveys a meaning; we associate it with the physical object. A Bostonian, a New Yorker, and a Tokyo commuter might each have a slightly different mental image—but the essence is similar. We have convenient shorthand notations for subway. In Boston, people refer to the subway as the “T.” If there is a “T” symbol with stairs going downward, that indicates a subway stop; if you see a “T” symbol on a street sign, with no stairs in sight, chances are that you’re at a bus stop. The symbol takes on a life of its own when a Bostonian says, “I’ll take the ‘T’ downtown.” On the other hand, New Yorkers look for an “M” (for Metropolitan Transit Authority) when they want to find a subway. But a New Yorker never says, “I’ll take the ‘M’ downtown.” To make matters muddier, in London a subway refers to an underground walkway, while the underground rail is referred to as “the tube.” Adding to the general zaniness of usage is a chain of “subway shops” selling submarine (hero, or grinder) sandwiches. And if you think you can clarify everything by switching to the term “underground railroad,” think again: Harriet Tubman’s underground railroad was something altogether different. Compared with this murky tangle, mathematical notation and usage may provide lucid relief. 4 CHAPTER 1 Functions Are Lurking Everywhere Exploratory Problems for Chapter 1 Calibrating Bottles From The Language of Functions and Graphs: An Examination Module for Secondary Schools, 1985, Shell Centre for Mathematical Education. 1.2 What Are Functions? Basic Vocabulary and Notation 5 1.2 WHAT ARE FUNCTIONS? BASIC VOCABULARY AND NOTATION ◆ EXAMPLE 1.1 The following table describes three possible designs for soda machines. Machine A Machine B Machine C Button # Output Button # Output Button # Output 1 Coke 1 Coke 1 Coke/Sprite 2 Diet Coke 2 Coke 2 Orange Crush/Diet Coke 3 Sprite 3 Diet Coke 3 Ginger Ale 4 Orange Crush 4 Diet Coke 4 Fresca 5 Ginger Ale 5 Coke 6 Coke You’ve seen machines similar to machines A and B, but you’ve probably never seen one similar to machine C. If you press button #1 of machine C you don’t know whether you’ll be getting a Coke or a Sprite; your input does not uniquely determine the output of the machine. ◆ Definitions A function is an input-output relationship with the characteristic that for each acceptable input there is exactly one corresponding output. In other words, the input completely determines the output. The input variable is referred to as the independent variable. The output generally depends on the input, and therefore the output variable is called the dependent variable. 2 The domain of a function is the set of all acceptable inputs. The range of a function is the set of all possible outputs. Functions are typically given short names, like f or g. 3 We typically call our generic function f (for function). For each input, x, in the domain of the function there is a corresponding output. This output is called the value of the function at x and is denoted f(x).Using set language, we say that a function f assigns to each element in its domain a specific element in its range. 2 We use the word “generally” because it is possible that all inputs result in the same output, i.e., that the function is a constant function. (A soda machine may be stocked only with Ginger Ale so that no matter what button you press—input— you’ll get the same output). 3 Some functions have longer names, like the cosine function, with the tag cos, or, to be more esoteric, the hyperbolic cosine function, with the tag cosh. 6 CHAPTER 1 Functions Are Lurking Everywhere Mental Picture: Function as a Machine input output Figure 1.2 You can think of a function as a machine (or box) that accepts an input and produces a predictable outcome. By “predictable outcome” we mean that if we input “button #2” into the machine and it produces a Diet Coke, we can predict, with certainty, that whenever we press button #2 the output will be a Diet Coke. Machine A can be modeled by a function because the button pressed (the input) com- pletely determines the soda received (the output). This machine has an additional feature—each button corresponds to a different soda. There is a one-to-one correspon- dence between inputs and outputs. A function having this characteristic is called 1-to-1. Notice that for machine A if you know the type of soda (the output), you know precisely which button was pressed (the corresponding input). Machine B can be modeled by a function because the button pressed (the input) completely determines the soda received (the output).The fact that more than one button will give the same type of soda does not prevent this relationship from being a function; it simply means that the function is not 1-to-1. When using machine B, if you know the type of soda (the output), you cannot determine which button was pressed. Machine C cannot be represented by a function because some buttons do not give a unique output. Selecting button #1 sometimes results in a can of Coke and other times in a can of Sprite. The input does not uniquely determine the output. Let’s make diagrammatic models of the three machines. We’ll call the mappings of inputs to outputs A, B, and C, corresponding to machines A, B, and C, respectively. Assign button #1 the number 1, button #2 the number 2, etc. We’ll use the first letter (small case) of each type of soda to represent the soda type, a convention that works because no two sodas listed begin with the same letter. Then we can represent the input-output relationships (mappings) as follows: AB C 1 −→ c 1 −→ c 1 −→ c or s 2 −→ d 2 −→ c 2 −→ o or d 3 −→ s 3 −→ d 3 −→ g 4 −→ o 4 −→ d 4 −→ f 5 −→ g 5 −→ c 6 −→ c 1.2 What Are Functions? Basic Vocabulary and Notation 7 A is a function with domain {1, 2, 3, 4, 5} and range {c, d, g, o, s}. 4 B is a function with domain {1, 2, 3, 4, 5, 6} and range {c, d}. C is not a function. To emphasize that mapping A is 1-to-1, mapping B is a function but not 1-to-1, and mapping C is not a function, we can represent the maps as follows: A B C 1c 1 c 1c 2d 2 s 3s 3 d 2o 4o 4 d 5g 5 3g 64f Mental Picture: Function as a mapping: assignment as arrows or but not Figure 1.3 You can think of a function as a mapping or association rule with no “split arrows” allowed. Multiple arrows can hit the same target; this simply indicates that the function is not 1-to-1. EXERCISE 1.1 Which of the following rules is a function? Are any of the functions 1-to-1? QR S 0 −→ 13 0 −→ π 0 −→ 0 1 −→ 14 1 −→ π 1 −→ ± 1 3 −→ 16 3 −→ π 3 −→ ± 3 6 −→ 11 6 −→ π 6 −→ ± 8 Answer Q and R are both functions, since every number in the domain has a unique output assigned to it; for each input there is one and only one output. S is not a function, because the output is not uniquely determined by the input. Q is 1-to-1; R is not 1-to-1. CAVEAT It is perfectly fine for many different inputs of a function to be assigned the same output. In fact, all inputs may be assigned to the same output, as illustrated by mapping R in the previous exercise. The “every-student-is-an-honors-student” rule, H , that assigns 4 Curly brackets, {},are used to enclose items in a set. 8 CHAPTER 1 Functions Are Lurking Everywhere to each student in our class the grade of “A” is a function. H is a function because H associates with each input (student) a single output (grade). Like the mapping R in the previous exercise, it is a constant function. Likewise, if the Red Queen’s procedure after her tea party is to assign to every guest the “off-with-his-head” command, her procedure can be modeled by a constant function, albeit a grisly one. A constant function is a function whose range consists of only one element. The output is fixed; it does not vary with the input. The structural aspect of an input-output relationship that makes it a function is that each input is associated with exactly one output. More casually, we could say that the input completely determines the output. By the phrase “completely determines” we are not implying causality (i.e., that the input causes the output to occur), nor are we implying a discernible or predictable pattern. (While we generally look for patterns, sometimes the phenomenon we are examining exhibits no pattern.) For example, let’s return to drink machine A, but this time cover up the key that tells us which buttons correspond to which drinks. Suppose we’ve never been to this machine before. We put in our money and press a button; we obtain a drink as output. Were we to put in the same amount of money and push the same button, we would get the same output. Before seeing the output for the first time we don’t know what it will be—but we do know that it is completely determined by the input. Although we cannot predict the output before using the machine, the machine operates like a function. Here the key is hidden. Now consider the input-output relationship that takes as input a date (day, month, year) and gives as output the highest temperature on the top of the Prudential Center, a skyscraper in downtown Boston. Is it a function? There is no causality between the date and the temperature, nor is there a formula to determine the temperature from the date, and yet, because each date is associated with exactly one high temperature, this is a function. We say that the temperature is a function of the date. Conversely, the date is not a function of the temperature. Why not? Functional Notation We use a very efficient shorthand notation to give information about the input-output relationship of a function. Let’s work with the function Q from Exercise 1.1. Suppose we want to indicate that the function Q assigns to the input of 3 the output of 16. We write Q(3) = 16 name of function input output We read this aloud as “Q of3is16.” 5 5 You can think of the notation Q(3) = 16 using the mental model of a function machine. Q is the function machine that has received the input 3 and spits out the output 16. To be anthropomorphic about it, think of the set of parentheses ( ) as a pair of hands cupped to receive the input. The output spurts out of the equal sign. 1.2 What Are Functions? Basic Vocabulary and Notation 9 Definition The output of a function f for a particular input is called the value of f at that input. The following statements are equivalent. The function f associates with the input 2 the output 7. The value of f at2is7. 2 f → 7 f(2)=7 The expression y = f(x)is shorthand for “y is the output of f corresponding to an input of x” and it is read “y equals f of x.” When we say y = f(x),wemean that y depends on x and is uniquely determined by x. ◆ EXAMPLE 1.2 Functional notation and the associated meaning. The Bee Line Trail in the White Moun- tains of New Hampshire is a hiking trail that goes directly up to the summit of Mount Chocorua. Let T(a)give the temperature (in degrees Fahrenheit) as a function of the alti- tude a (in feet above sea level) at a particular time on a particular day along the Bee Line Trail. altitude T −→ temperature Suppose that right now you are at an altitude of H feet above sea level and that temperature is a function of altitude. Answer the questions that follow. Q: What is the temperature at your altitude? A: Use your altitude, H , as the input of the function T ; T(H)is the temperature. Q: What is the temperature if you double your current altitude? A: Use the input 2H , twice your current altitude; the temperature will be T(2H). Q: What is the temperature if you double your original altitude and then ascend 1000 more feet up the mountain? A: Use the input 2H + 1000, which is 1000 feet above an altitude of 2H ; the temperature will be T(2H +1000). Q: Interpret the meanings of T H 2 and T(H) 2 . A: T H 2 is the temperature at one-half of your current altitude; the input (the altitude) has been halved. T(H) 2 is one-half of the temperature at your current altitude; the output (temperature) has been halved. Notice that in thefirst expression the new altitudeis specified, and in the second expression the new temperature is specified. Also note that the two expressions will generally not be equal. If it is 60 degrees at 2000 feet, then it probably won’t be 30 degrees at 1000 feet. Q: Interpret T(H +10) and T(H)+10. A: T(H +10) is the temperature at an altitude 10 feet higher than your current altitude. T(H)+10 is a temperature 10 degrees higher than your current temperature. Notice that 10 CHAPTER 1 Functions Are Lurking Everywhere the 10 inside the parentheses represents 10 more feet while the 10 outside the parentheses represents 10 more degrees. Q: At what altitude will the temperature be 40 degrees? A: Solve the equation T(a)= 40 for a. (There may not be a value for a that satisfies this equation. At the moment in question it is possible that at no height on the mountain is the temperature 40 degrees.) Q: How high must you go for the temperature to drop 10 degrees? A: The temperature at your current height is T(H)degrees, thus we need to find the altitude that gives a temperature of T(H)−10 degrees. This means solving for a in the equation T(a)= T(H)−10. (Note: H is a constant and a is the variable.) ◆ PROBLEMS FOR SECTION 1.2 1. Below are pictures of three different train trips. In each case, the train is far enough from its station of origin to have reached a steady cruising speed. One picture corresponds to a trip on Amtrak in America; the second corresponds to a train trip in the People’s Republic of China, where trains have a tendency to be slow and train travel is a very leisurely affair; and the third corresponds to a trip on the Japanese bullet train (shinkansen), a very fast train. Which picture corresponds to which trip? distance from station C A B time 2. Which of the following rules can be modeled as a function? If a rule is not a function, explain why not. (a) For a particular flask, the rule assigns to every volume (input) of liquid in the flask the corresponding height (output). (b) For a particular flask, the rule assigns to every height (input) the corresponding volume (output). (c) The rule assigns to every person his or her birthday. (d) The rule assigns to every recorded singer the title of his or her first recorded song. (e) The rule assigns to every state the current representative in the House of Repre- sentatives. (f) The rule assigns to every current member of the House of Representatives the state he or she represents. . we want to indicate that the function Q assigns to the input of 3 the output of 16. We write Q (3) = 16 name of function input output We read this aloud as “Q of3 is16.” 5 5 You can think of the. output of f corresponding to an input of x” and it is read “y equals f of x.” When we say y = f(x),wemean that y depends on x and is uniquely determined by x. ◆ EXAMPLE 1.2 Functional notation and. −→ ± 1 3 −→ 16 3 −→ π 3 −→ ± 3 6 −→ 11 6 −→ π 6 −→ ± 8 Answer Q and R are both functions, since every number in the domain has a unique output assigned to it; for each input there is one and only