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0.1 FUNCTIONS
In this section we will define and develop the concept of a “function,” which is the basic mathematical object that scientists and mathematicians use to describe relationships between variable quantities. Functions play a central role in calculus and its applications.
DEFINITION OF A FUNCTION
Many scientific laws and engineering principles describe how one quantity depends on another. This idea was formalized in 1673 by Gottfried Wilhelm Leibniz (see p. xx) who coined the term function to indicate the dependence of one quantity on another, as described in the following definition.
0.1.1 DEFINITION
If a variable y depends on a variable x in such a way that each value of x determines exactly one value of y , then we say that y is a function of x. Four common methods for representing functions are:
- Numerically by tables - Algebraically by formulas - Geometrically by graphs - Verbally The method of representation often depends on how the function arises. For example: - Table 0.1.1 shows the top qualifying speed S for the Indianapolis 500 auto race as a function of the year t. There is exactly one value of S for each value of t. - Figure 0.1.1 is a graphical record of an earthquake recorded on a seismograph. The graph describes the deflection D of the seismograph needle as a function of the time T elapsed since the wave left the earthquake's epicenter. There is exactly one value of D for each value of T. - Some of the most familiar functions arise from formulas; for example, the formula expresses the circumference C of a circle as a function of its radius r. There is exactly one value of C for each value of r. - Sometimes functions are described in words. For example, Isaac Newton's Law of Universal Gravitation is often stated as follows: The gravitational force of attraction between two bodies in the Universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This is the verbal description of the formula in which F is the force of attraction, and are the masses, r is the distance between them, and G is a constant. If the masses are constant, then the verbal description defines F as a function of r. There is exactly one value of F for each value of r. Table 0.1.
Figure 0.1. In the mid-eighteenth century the Swiss mathematician Leonhard Euler (pronounced “oiler”) conceived the idea of denoting functions by letters of the alphabet, thereby making it possible to refer to functions without stating specific formulas, graphs, or tables. To understand Euler's idea, think of a function as a computer program that takes an input x , operates on it in some way, and produces exactly one output y. The computer program is an object in its own right, so we can give it a name, say f. Thus, the function f (the computer program) associates a unique output y with each input x (Figure 0.1.2). This suggests the following definition. 0.1.2 DEFINITION A function f is a rule that associates a unique output with each input. If the input is denoted by x , then the output is denoted by (read “ f of x ”). Figure 0.1. In this definition the term unique means “exactly one.” Thus, a function cannot assign two different outputs to the same input. For example, Figure 0.1.3 shows a plot of weight versus age for a random sample of 100 college students. This plot does not describe W as a function of A because there are some values of A with more than one corresponding value of W. This is to be expected, since two people with the same age can have different weights.
problems of lunar motion that baffled Isaac Newton and once did a complicated calculation in his head to settle an argument between two students whose computations differed in the fiftieth decimal place. Following the development of calculus by Leibniz and Newton, results in mathematics developed rapidly in a disorganized way. Euler's genius gave coherence to the mathematical landscape. He was the first mathematician to bring the full power of calculus to bear on problems from physics. He made major contributions to virtually every branch of mathematics as well as to the theory of optics, planetary motion, electricity, magnetism, and general mechanics.
GRAPHS OF FUNCTIONS
If f is a real-valued function of a real variable, then the graph of f in the xy -plane is defined to be the graph of the equation. For example, the graph of the function is the graph of the equation , shown in Figure 0.1.4. That figure also shows the graphs of some other basic functions that may already be familiar to you. In Appendix A we discuss techniques for graphing functions using graphing technology. Figure 0.1. Figure 0.1.4 shows only portions of the graphs. Where appropriate, and unless indicated otherwise, it is understood that graphs shown in this text extend indefinitely beyond the boundaries of the displayed figure. Since is imaginary for negative values of x , there are no points on the graph of in the region where. Graphs can provide valuable visual information about a function. For example, since the graph of a function f in the xy -plane is the graph of the equation , the points on the graph of f are of the form ; that is, the y-coordinate of a point on the graph of f is the value of f at the corresponding x-coordinate (Figure 0.1.5). The values of x for which are the x -coordinates of the points where the graph of f intersects the x -axis (Figure 0.1.6). These values are called the zeros of f , the roots of , or the x-intercepts of the graph of. Figure 0.1.5 The y -coordinate of a point on the graph of is the value of f at the corresponding x -coordinate.
Figure 0.1.6 f has zeros at.
THE VERTICAL LINE TEST
Not every curve in the xy -plane is the graph of a function. For example, consider the curve in Figure 0.1.7, which is cut at two distinct points, and , by a vertical line. This curve cannot be the graph of for any function f ; otherwise, we would have which is impossible, since f cannot assign two different values to a. Thus, there is no function f whose graph is the given curve. This illustrates the following general result, which we will call the vertical line test. Figure 0.1.7 This curve cannot be the graph of a function. 0.1.3 THE VERTICAL LINE TEST A curve in the xy -plane is the graph of some function f if and only if no vertical line intersects the curve more than once.
Example 3
The graph of the equation is a circle of radius 5 centered at the origin and hence there are vertical lines that cut the graph more than once (Figure 0.1.8). Thus this equation does not define y as a function of x. Figure 0.1.
THE ABSOLUTE VALUE FUNCTION
Recall that the absolute value or magnitude of a real number x is defined by
TECHNOLOGY MASTERY
Verify (1) by using a graphing utility to show that the equations and have the same graph.
PIECEWISE-DEFINED FUNCTIONS
The absolute value function is an example of a function that is defined piecewise in the sense that the formula for f changes, depending on the value of x.
Example 4
Sketch the graph of the function defined piecewise by the formula
Solution. The formula for f changes at the points and. (We call these the
breakpoints for the formula.) A good procedure for graphing functions defined piecewise is to graph the function separately over the open intervals determined by the breakpoints, and then graph f at the breakpoints themselves. For the function f in this example the graph is the horizontal ray on the interval , it is the semicircle on the interval , and it is the ray on the interval. The formula for f specifies that the equation applies at the breakpoint [so ], and it specifies that the equation applies at the breakpoint 1 [so ]. The graph of f is shown in Figure 0.1.10. Figure 0.1. REMARK In Figure 0.1.10 the solid dot and open circle at the breakpoint serve to emphasize that the point on the graph lies on the ray and not the semicircle. There is no ambiguity at the breakpoint because the two parts of the graph join together continuously there.
Example 5
Increasing the speed at which air moves over a person's skin increases the rate of moisture evaporation and makes the person feel cooler. (This is why we fan ourselves in hot weather.) The wind chill index is the temperature at a wind speed of 4 mi/h that would produce the same sensation on exposed skin as the current temperature and wind speed combination. An empirical formula (i.e., a formula based on experimental data) for the wind chill index W at 32° F for a wind speed of v mi/h is A computer-generated graph of is shown in Figure 0.1.11.
Figure 0.1.11 Wind chill versus wind speed at 32°F The wind chill index measures the sensation of coldness that we feel from the combined effect of temperature and wind speed.
DOMAIN AND RANGE
If x and y are related by the equation , then the set of all allowable inputs ( x -values) is called the domain of f , and the set of outputs ( y -values) that result when x varies over the domain is called the range of f. For example, if f is the function defined by the table in Example 1, then the domain is the set and the range is the set. Sometimes physical or geometric considerations impose restrictions on the allowable inputs of a function. For example, if y denotes the area of a square of side x , then these variables are related by the equation. Although this equation produces a unique value of y for every real number x , the fact that lengths must be nonnegative imposes the requirement that. One might argue that a physical square cannot have a side of length zero. However, it is often convenient mathematically to allow zero lengths, and we will do so throughout this text where appropriate. When a function is defined by a mathematical formula, the formula itself may impose restrictions on the allowable inputs. For example, if , then is not an allowable input since division by zero is undefined, and if , then negative values of x are not allowable inputs because they produce imaginary values for y and we have agreed to consider only real-valued functions of a real variable. In general, we make the following definition. 0.1.5 DEFINITION If a real-valued function of a real variable is defined by a formula, and if no domain is stated explicitly, then it is to be understood that the domain consists of all real numbers for which the formula yields a real value. This is called the natural domain of the function.
Figure 0.1.
THE EFFECT OF ALGEBRAIC OPERATIONS ON THE DOMAIN
Algebraic expressions are frequently simplified by canceling common factors in the numerator and denominator. However, care must be exercised when simplifying formulas for functions in this way, since this process can alter the domain.
Example 7
The natural domain of the function (2) consists of all real x except. However, if we factor the numerator and then cancel the common factor in the numerator and denominator, we obtain (3) Since the right side of (3) has a value of and was undefined in (2), the algebraic simplification has changed the function. Geometrically, the graph of (3) is the line in Figure 0.1.14 a , whereas the graph of (2) is the same line but with a hole at , since the function is undefined there (Figure 0.1.14 b ). In short, the geometric effect of the algebraic cancellation is to eliminate the hole in the original graph. Figure 0.1. Sometimes alterations to the domain of a function that result from algebraic simplification are irrelevant to the problem at hand and can be ignored. However, if the domain must be preserved, then one must impose the restrictions on the simplified function explicitly. For example, if we wanted to preserve the domain of the
function in Example 7, then we would have to express the simplified form of the function as
Example 8
Find the domain and range of (a) (b)
Solution.
(a) Since no domain is stated explicitly, the domain of f is its natural domain,. As x varies over the interval , the value of varies over the interval , so the value of varies over the interval , which is the range of f. The domain and range are highlighted in green on the x - and y -axes in Figure 0.1.15. Figure 0.1. (b) The given function f is defined for all real x , except , so the natural domain of f is To determine the range it will be convenient to introduce a dependent variable (4) Although the set of possible y -values is not immediately evident from this equation, the graph of (4), which is shown in Figure 0.1.16, suggests that the range of f consists of all y , except. To see that this is so, we solve (4) for x in terms of y : Figure 0.1. It is now evident from the right side of this equation that is not in the range; otherwise we would have a division by zero. No other values of y are excluded by this equation, so the range of the function f is , which agrees with the result obtained graphically.
DOMAIN AND RANGE IN APPLIED PROBLEMS
In applications, physical considerations often impose restrictions on the domain and range of a function.
Example 9
In geometric problems where you want to preserve the “true” shape of a graph, you must use units of equal length on both axes. For example, if you graph a circle in a coordinate system in which 1 unit in the y - direction is smaller than 1 unit in the x -direction, then the circle will be squashed vertically into an elliptical shape (Figure 0.1.19). Figure 0.1. In applications where the variables on the two axes have unrelated units (say, centimeters on the y -axis and seconds on the x -axis), then nothing is gained by requiring the units to have equal lengths; choose the lengths to make the graph as clear as possible. However, sometimes it is inconvenient or impossible to display a graph using units of equal length. For example, consider the equation If we want to show the portion of the graph over the interval , then there is no problem using units of equal length, since y only varies from 0 to 9 over that interval. However, if we want to show the portion of the graph over the interval , then there is a problem keeping the units equal in length, since the value of y varies between 0 and 100. In this case the only reasonable way to show all of the graph that occurs over the interval is to compress the unit of length along the y -axis, as illustrated in Figure 0.1.20. Figure 0.1.
QUICK CHECK EXERCISES 0.
1. (^) Let. (a) The natural domain of f is _________. (b) _________ (c) (^) _________ (d) if _________ (e) The range of f is _________. Answer: (a) (b) 6 (c) (d) 8 (e)
2. Line segments in xy -plane form “letters” as depicted. (a) If the y -axis is parallel to the letter I, which of the letters represent the graph of for some function f? (b) If the y -axis is perpendicular to the letter I, which of the letters represent the graph of for some function f? Answer: (a) M (b) I 3. The accompanying figure shows the complete graph of. (a) The domain of f is _________. (b) The range of f is _________. (c) _________ (d) (^) _________ (e) (^) The solutions to are _________ and _________. Figure Ex- Answer: (a) (b) (c) (d) 1 (e) (^) ; 4. The accompanying table gives a 5-day forecast of high and low temperatures in degrees Fahrenheit (°F). (a) Suppose that x and y denote the respective high and low temperature predictions for each of the 5 days. Is y a function of x? If so, give the domain and range of this function. (b) Suppose that x and y denote the respective low and high temperature predictions for each of the 5 days. Is y a function of x? If so, give the domain and range of this function. Table Ex- Answer: (a) yes; domain: range: (b) no 5. Let l , w , and A denote the length, width, and area of a rectangle, respectively, and suppose that the width of the rectangle is half the length. (a) If l is expressed as a function of w , then _________. (b) If A is expressed as a function of l , then _________. (c) If w is expressed as a function of A , then _________. Answer:
FOCUS ON CONCEPTS
5. The accompanying graph shows the median income in U.S. households (adjusted for inflation) between 1990 and 2005. Use the graph to answer the following questions, making reasonable approximations where needed. (a) When was the median income at its maximum value, and what was the median income when that occurred? (b) When was the median income at its minimum value, and what was the median income when that occurred? (c) The median income was declining during the 2-year period between 2000 and 2002. Was it declining more rapidly during the first year or the second year of that period? Explain your reasoning. Figure Ex- 6. Use the median income graph in Exercise 5 to answer the following questions, making reasonable approximations where needed. (a) What was the average yearly growth of median income between 1993 and 1999? (b) The median income was increasing during the 6-year period between 1993 and 1999. Was it increasing more rapidly during the first 3 years or the last 3 years of that period? Explain your reasoning. (c) Consider the statement: “After years of decline, median income this year was finally higher than that of last year.” In what years would this statement have been correct? 7. (^) Find , and. (a) (b) 8. (^) Find , and. (a) (b) 9–10. Find the natural domain and determine the range of each function. If you have a graphing utility, use it to confirm that your result is consistent with the graph produced by your graphing utility. [ Note: Set your graphing utility in radian mode when graphing trigonometric functions.] 9. (^) (a) (b) (c) (d) (e) (f) 10. (^) (a)
(b) (c) (d) (e) (f)
FOCUS ON CONCEPTS
11. (^) (a) If you had a device that could record the Earth's population continuously, would you expect the graph of population versus time to be a continuous (unbroken) curve? Explain what might cause breaks in the curve. (b) Suppose that a hospital patient receives an injection of an antibiotic every 8 hours and that between injections the concentration C of the antibiotic in the bloodstream decreases as the antibiotic is absorbed by the tissues. What might the graph of C versus the elapsed time t look like? 12. (^) (a) If you had a device that could record the temperature of a room continuously over a 24-hour period, would you expect the graph of temperature versus time to be a continuous (unbroken) curve? Explain your reasoning. (b) If you had a computer that could track the number of boxes of cereal on the shelf of a market continuously over a 1-week period, would you expect the graph of the number of boxes on the shelf versus time to be a continuous (unbroken) curve? Explain your reasoning. 13. A boat is bobbing up and down on some gentle waves. Suddenly it gets hit by a large wave and sinks. Sketch a rough graph of the height of the boat above the ocean floor as a function of time. 14. A cup of hot coffee sits on a table. You pour in some cool milk and let it sit for an hour. Sketch a rough graph of the temperature of the coffee as a function of time. 15–18. As seen in Example 3, the equation does not define y as a function of x. Each graph in these exercises is a portion of the circle. In each case, determine whether the graph defines y as a function of x , and if so, give a formula for y in terms of x. **15.
18.**
boxes of maximum volume. Figure Ex-
30. Repeat Exercise 29 assuming the box is constructed in the same fashion from a 6-inch-square sheet of metal. 31. (^) A construction company has adjoined a 1000 ft^2 rectangular enclosure to its office building. Three sides of the enclosure are fenced in. The side of the building adjacent to the enclosure is 100 ft long and a portion of this side is used as the fourth side of the enclosure. Let x and y be the dimensions of the enclosure, where x is measured parallel to the building, and let L be the length of fencing required for those dimensions. (a) Find a formula for L in terms of x and y. (b) Find a formula that expresses L as a function of x alone. (c) What is the domain of the function in part (b)? (d) Plot the function in part (b) and estimate the dimensions of the enclosure that minimize the amount of fencing required. 32. As shown in the accompanying figure, a camera is mounted at a point 3000 ft from the base of a rocket launching pad. The rocket rises vertically when launched, and the camera's elevation angle is continually adjusted to follow the bottom of the rocket. (a) Express the height x as a function of the elevation angle. (b) Find the domain of the function in part (a). (c) Plot the graph of the function in part (a) and use it to estimate the height of the rocket when the elevation angle is radian. Compare this estimate to the exact height. Figure Ex- 33. A soup company wants to manufacture a can in the shape of a right circular cylinder that will hold 500 cm^3 of liquid. The material for the top and bottom costs 0.02 cent/cm^2 , and the material for the sides costs 0.01 cent/cm^2. (a) Estimate the radius r and the height h of the can that costs the least to manufacture. [ Suggestion: Express the cost C in terms of r .] (b) Suppose that the tops and bottoms of radius r are punched out from square sheets with sides of length and the scraps are waste. If you allow for the cost of the waste, would you expect the can of least cost to be taller or shorter than the one in part (a)? Explain. (c) Estimate the radius, height, and cost of the can in part (b), and determine whether your conjecture was correct. 34. The designer of a sports facility wants to put a quarter-mile (1320 ft) running track around a football field, oriented as in the accompanying figure. The football field is 360 ft long (including the end zones) and 160 ft wide. The track consists of two straightaways and two semicircles, with the straightaways extending at least the length of the football field. (a) Show that it is possible to construct a quarter-mile track around the football field. [ Suggestion: Find the shortest track that can be constructed around the field.] (b) Let L be the length of a straightaway (in feet), and let x be the distance (in feet) between a sideline of the football field and a straightaway. Make a graph of L versus x. (c) Use the graph to estimate the value of x that produces the shortest straightaways, and then find this value of x exactly. (d) Use the graph to estimate the length of the longest possible straightaways, and then find that length exactly.
Figure Ex- 35–36. (i) Explain why the function f has one or more holes in its graph, and state the x -values at which those holes occur. (ii) Find a function g whose graph is identical to that of f , but without the holes. 35. 36.
37. In 2001 the National Weather Service introduced a new wind chill temperature (WCT) index. For a given outside temperature T and wind speed v , the wind chill temperature index is the equivalent temperature that exposed skin would feel with a wind speed of v mi/h. Based on a more accurate model of cooling due to wind, the new formula is where T is the temperature in ° F, v is the wind speed in mi/h, and WCT is the equivalent temperature in ° F. Find the WCT to the nearest degree if and (a) (b) (c) Source: Adapted from UMAP Module 658, Windchill , W. Bosch and L. Cobb, COMAP, Arlington, MA. 38–40. Use the formula for the wind chill temperature index described in Exercise 37. 38. Find the air temperature to the nearest degree if the WCT is reported as with a wind speed of 48 mi/h. 39. Find the air temperature to the nearest degree if the WCT is reported as with a wind speed of 48 mi/h. 40. Find the wind speed to the nearest mile per hour if the WCT is reported as with an air temperature of. Copyright © 2009 John Wiley & Sons, Inc. All rights reserved.
