Functions. Introduction CHAPTER OUTLINE

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1 Functions,00 P, Figure Standard and Poor s Inde with dividends reinvested (credit "bull": modification of work b Praitno Hadinata; credit "graph": modification of work b MeasuringWorth) CHAPTER OUTLINE. Functions and Function Notation. Domain and Range. Rates of Change and Behavior of Graphs. Composition of Functions. Transformation of Functions.6 Absolute Value Functions.7 Inverse Functions Introduction Toward the end of the twentieth centur, the values of stocks of Internet and technolog companies rose dramaticall. As a result, the Standard and Poor s stock market average rose as well. Figure tracks the value of that initial investment of just under $00 over the 0 ears. It shows that an investment that was worth less than $00 until about 99 skrocketed up to about $,00 b the beginning of 000. That five-ear period became known as the dot-com bubble because so man Internet startups were formed. As bubbles tend to do, though, the dot-com bubble eventuall burst. Man companies grew too fast and then suddenl went out of business. The result caused the sharp decline represented on the graph beginning at the end of 000. Notice, as we consider this eample, that there is a definite relationship between the ear and stock market average. For an ear we choose, we can determine the corresponding value of the stock market average. In this chapter, we will eplore these kinds of relationships and their properties. Download for free at 9

2 96 CHAPTER FUNCTIONS LEARNING OBJECTIVES In this section, ou will: Find the average rate of change of a function. Use a graph to determine where a function is increasing, decreasing, or constant. Use a graph to locate local maima and local minima. Use a graph to locate the absolute maimum and absolute minimum.. RATES OF CHANGE AND BEHAVIOR OF GRAPHS Gasoline costs have eperienced some wild fluctuations over the last several decades. Table [] lists the average cost, in dollars, of a gallon of gasoline for the ears The cost of gasoline can be considered as a function of ear C() Table If we were interested onl in how the gasoline prices changed between 00 and 0, we could compute that the cost per gallon had increased from $. to $.68, an increase of $.7. While this is interesting, it might be more useful to look at how much the price changed per ear. In this section, we will investigate changes such as these. Finding the Average Rate of Change of a Function The price change per ear is a rate of change because it describes how an output quantit changes relative to the change in the input quantit. We can see that the price of gasoline in Table did not change b the same amount each ear, so the rate of change was not constant. If we use onl the beginning and ending data, we would be finding the average rate of change over the specified period of time. To find the average rate of change, we divide the change in the output value b the change in the input value. Average rate of change Change in output Change in input _ _ f ( ) f ( ) The Greek letter (delta) signifies the change in a quantit; we read the ratio as delta- over delta- or the change in divided b the change in. Occasionall we write f instead of, which still represents the change in the function s output value resulting from a change to its input value. It does not mean we are changing the function into some other function. In our eample, the gasoline price increased b $.7 from 00 to 0. Over 7 ears, the average rate of change was _ _ $ dollars per ear 7 ears On average, the price of gas increased b about 9.6 each ear. Other eamples of rates of change include: A population of rats increasing b 0 rats per week A car traveling 68 miles per hour (distance traveled changes b 68 miles each hour as time passes) A car driving 7 miles per gallon (distance traveled changes b 7 miles for each gallon) The current through an electrical circuit increasing b 0. amperes for ever volt of increased voltage The amount of mone in a college account decreasing b $,000 per quarter Accessed //0. Download for free at

3 SECTION. RATES OF CHANGE AND BEHAVIOR OF GRAPHS 97 rate of change A rate of change describes how an output quantit changes relative to the change in the input quantit. The units on a rate of change are output units per input units. The average rate of change between two input values is the total change of the function values (output values) divided b the change in the input values. _ f ( ) f ( ) How To Given the value of a function at different points, calculate the average rate of change of a function for the interval between two values and.. Calculate the difference.. Calculate the difference.. Find the ratio _. Eample Computing an Average Rate of Change Using the data in Table, find the average rate of change of the price of gasoline between 007 and 009. Solution In 007, the price of gasoline was $.8. In 009, the cost was $.. The average rate of change is _ $. $ $0. ears $0. per ear Analsis Note that a decrease is epressed b a negative change or negative increase. A rate of change is negative when the output decreases as the input increases or when the output increases as the input decreases. Tr It # Using the data in Table at the beginning of this section, find the average rate of change between 00 and 00. Eample Computing Average Rate of Change from a Graph Given the function g (t) shown in Figure, find the average rate of change on the interval [, ]. g(t) t Figure Download for free at

4 98 CHAPTER FUNCTIONS Solution At t, Figure shows g ( ). At t, the graph shows g (). g(t) g(t) t t Figure The horizontal change t is shown b the red arrow, and the vertical change g(t) is shown b the turquoise arrow. The average rate of change is shown b the slope of the red line segment. The output changes b while the input changes b, giving an average rate of change of ( ) Analsis Note that the order we choose is ver important. If, for eample, we use _, we will not get the correct answer. Decide which point will be and which point will be, and keep the coordinates fied as (, ) and (, ). Eample Computing Average Rate of Change from a Table After picking up a friend who lives 0 miles awa and leaving on a trip, Anna records her distance from home over time. The values are shown in Table. Find her average speed over the first 6 hours. t (hours) D(t) (miles) Table Solution Here, the average speed is the average rate of change. She traveled 8 miles in 6 hours. The average speed is 7 miles per hour Analsis Because the speed is not constant, the average speed depends on the interval chosen. For the interval [, ], the average speed is 6 miles per hour. Eample Computing Average Rate of Change for a Function Epressed as a Formula Compute the average rate of change of f () on the interval [, ]. Solution We can start b computing the function values at each endpoint of the interval. f () 7 Now we compute the average rate of change. f() 6 6 Download for free at

5 SECTION. RATES OF CHANGE AND BEHAVIOR OF GRAPHS 99 Tr It # Average rate of change f () f () _ Find the average rate of change of f () on the interval [, 9]. Eample Finding the Average Rate of Change of a Force The electrostatic force F, measured in newtons, between two charged particles can be related to the distance between the particles d, in centimeters, b the formula F(d) _. Find the average rate of change of force if the distance d between the particles is increased from cm to 6 cm. Solution We are computing the average rate of change of F (d) on the interval [, 6]. d Average rate of change F(6) F() 6 _ 6 6 Simplif _ 9 The average rate of change is newton per centimeter. 9 Eample 6 _ Finding an Average Rate of Change as an Epression Combine numerator terms. Simplif. Find the average rate of change of g(t) t + t + on the interval [0, a]. The answer will be an epression involving a in simplest form. Solution We use the average rate of change formula. Average rate of change g(a) g(0) _ a 0 (a + a + ) (0 + (0) + ) a 0 a + a + a a(a + ) a a + Evaluate. Simplif. Simplif and factor. Divide b the common factor a. This result tells us the average rate of change in terms of a between t 0 and an other point t a. For eample, on the interval [0, ], the average rate of change would be + 8. Download for free at

6 00 CHAPTER FUNCTIONS Tr It # Find the average rate of change of f () + 8 on the interval [, a] in simplest forms in terms of a. Using a Graph to Determine Where a Function is, Decreasing, or Constant As part of eploring how functions change, we can identif intervals over which the function is changing in specific was. We sa that a function is increasing on an interval if the function values increase as the input values increase within that interval. Similarl, a function is decreasing on an interval if the function values decrease as the input values increase over that interval. The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative. Figure shows eamples of increasing and decreasing intervals on a function. f() Decreasing f(b) > f(a) where b > a f(b) < f(a) where b > a f(b) > f(a) where b > a Figure The function f() is increasing on (, ) (, ) and is decreasing on (, ). While some functions are increasing (or decreasing) over their entire domain, man others are not. A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a local maimum. If a function has more than one, we sa it has local maima. Similarl, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a local minimum. The plural form is local minima. Together, local maima and minima are called local etrema, or local etreme values, of the function. (The singular form is etremum. ) Often, the term local is replaced b the term relative. In this tet, we will use the term local. Clearl, a function is neither increasing nor decreasing on an interval where it is constant. A function is also neither increasing nor decreasing at etrema. Note that we have to speak of local etrema, because an given local etremum as defined here is not necessaril the highest maimum or lowest minimum in the function s entire domain. For the function whose graph is shown in Figure, the local maimum is 6, and it occurs at. The local minimum is 6 and it occurs at. Local maimum 6 occurs at f() (, 6) Decreasing (, 6) Local minimum 6 occurs at Figure Download for free at

7 SECTION. RATES OF CHANGE AND BEHAVIOR OF GRAPHS 0 To locate the local maima and minima from a graph, we need to observe the graph to determine where the graph attains its highest and lowest points, respectivel, within an open interval. Like the summit of a roller coaster, the graph of a function is higher at a local maimum than at nearb points on both sides. The graph will also be lower at a local minimum than at neighboring points. Figure illustrates these ideas for a local maimum. f() f(b) Local maimum a function b Decreasing function Figure Definition of a local maimum c These observations lead us to a formal definition of local etrema. local minima and local maima A function f is an increasing function on an open interval if f (b) > f (a) for an two input values a and b in the given interval where b > a. A function f is a decreasing function on an open interval if f (b) < f (a) for an two input values a and b in the given interval where b > a. A function f has a local maimum at b if there eists an interval (a, c) with a < b < c such that, for an in the interval (a, c), f () f (b). Likewise, f has a local minimum at b if there eists an interval (a, c) with a < b < c such that, for an in the interval (a, c), f () f (b). Eample 7 Finding and Decreasing Intervals on a Graph Given the function p(t) in Figure 6, identif the intervals on which the function appears to be increasing. p 6 t Figure 6 Solution We see that the function is not constant on an interval. The function is increasing where it slants upward as we move to the right and decreasing where it slants downward as we move to the right. The function appears to be increasing from t to t and from t on. In interval notation, we would sa the function appears to be increasing on the interval (, ) and the interval (, ). Download for free at

8 0 CHAPTER FUNCTIONS Analsis Notice in this eample that we used open intervals (intervals that do not include the endpoints), because the function is neither increasing nor decreasing at t, t, and t. These points are the local etrema (two minima and a maimum). Eample 8 Finding Local Etrema from a Graph Graph the function f () +. Then use the graph to estimate the local etrema of the function and to determine the intervals on which the function is increasing. Solution Using technolog, we find that the graph of the function looks like that in Figure 7. It appears there is a low point, or local minimum, between and, and a mirror-image high point, or local maimum, somewhere between and. f() Figure 7 Analsis Most graphing calculators and graphing utilities can estimate the location of maima and minima. Figure 8 provides screen images from two different technologies, showing the estimate for the local maimum and minimum ,.699 (a) 6 Maimum X.99 Y.699 Figure 8 Based on these estimates, the function is increasing on the interval (,.9) and (.9, ). Notice that, while we epect the etrema to be smmetric, the two different technologies agree onl up to four decimals due to the differing approimation algorithms used b each. (The eact location of the etrema is at ± 6, but determining this requires calculus.) (b) Tr It # Graph the function f () to estimate the local etrema of the function. Use these to determine the intervals on which the function is increasing and decreasing. Download for free at

9 SECTION. RATES OF CHANGE AND BEHAVIOR OF GRAPHS 0 Eample 9 Finding Local Maima and Minima from a Graph For the function f whose graph is shown in Figure 9, find all local maima and minima f Figure 9 Solution Observe the graph of f. The graph attains a local maimum at because it is the highest point in an open interval around. The local maimum is the -coordinate at, which is. The graph attains a local minimum at because it is the lowest point in an open interval around. The local minimum is the -coordinate at, which is. Analzing the Toolkit Functions for or Decreasing Intervals We will now return to our toolkit functions and discuss their graphical behavior in Figure 0, Figure, and Figure. Function /Decreasing Eample Constant Function f() c Neither increasing nor decreasing Identit Function f() Quadratic Function f() on (0, ) Decreasing on (, 0) Minimum at 0 Figure 0 Download for free at

10 0 CHAPTER FUNCTIONS Function /Decreasing Eample Cubic Function f() Reciprocal f() Decreasing (, 0) (0, ) Reciprocal Squared f() _ on (, 0) Decreasing on (0, ) Figure Function /Decreasing Eample Cube Root f() Square Root f() on (0, ) Absolute Value f() on (0, ) Decreasing on (, 0) Figure Download for free at

11 SECTION. RATES OF CHANGE AND BEHAVIOR OF GRAPHS 0 Use A Graph to Locate the Absolute Maimum and Absolute Minimum There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locall) and locating the highest and lowest points on the graph for the entire domain. The -coordinates (output) at the highest and lowest points are called the absolute maimum and absolute minimum, respectivel. To locate absolute maima and minima from a graph, we need to observe the graph to determine where the graph attains it highest and lowest points on the domain of the function. See Figure. f Absolute maimum is f () Absolute minimum is f (0) Figure Not ever function has an absolute maimum or minimum value. The toolkit function f () is one such function. absolute maima and minima The absolute maimum of f at c is f (c) where f (c) f () for all in the domain of f. The absolute minimum of f at d is f (d) where f (d) f () for all in the domain of f. Eample 0 Finding Absolute Maima and Minima from a Graph For the function f shown in Figure, find all absolute maima and minima f Figure Solution Observe the graph of f. The graph attains an absolute maimum in two locations, and, because at these locations, the graph attains its highest point on the domain of the function. The absolute maimum is the -coordinate at and, which is 6. The graph attains an absolute minimum at, because it is the lowest point on the domain of the function s graph. The absolute minimum is the -coordinate at, which is 0. Access this online resource for additional instruction and practice with rates of change. Average Rate of Change ( Download for free at

12 06 CHAPTER FUNCTIONS. SECTION EXERCISES VERBAL. Can the average rate of change of a function be constant?. How are the absolute maimum and minimum similar to and different from the local etrema?. If a function f is increasing on (a, b) and decreasing on (b, c), then what can be said about the local etremum of f on (a, c)?. How does the graph of the absolute value function compare to the graph of the quadratic function,, in terms of increasing and decreasing intervals? ALGEBRAIC For the following eercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.. f () 7 on [, b] 6. g () 9 on [, b] 7. p() + on [, + h] 8. k() on [, + h] 9. f () + on [, + h] 0. g() on [, + h]. a(t) t + on [9, 9 + h]. b() on [, + h] +. j() on [, + h]. r(t) t on [, + h]. f ( + h) f() given f() on [, + h] h GRAPHICAL For the following eercises, consider the graph of f shown in Figure. 6. Estimate the average rate of change from to Estimate the average rate of change from to Figure For the following eercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing Download for free at

13 SECTION. SECTION EXERCISES For the following eercises, consider the graph shown in Figure 6.. Estimate the intervals where the function is increasing or decreasing.. Estimate the point(s) at which the graph of f has a local maimum or a local minimum For the following eercises, consider the graph in Figure 7. Figure 6. If the complete graph of the function is shown, estimate the intervals where the function is increasing or decreasing.. If the complete graph of the function is shown, estimate the absolute maimum and absolute minimum Figure 7 NUMERIC 6. Table gives the annual sales (in millions of dollars) of a product from 998 to 006. What was the average rate of change of annual sales (a) between 00 and 00, and (b) between 00 and 00? Year Sales (millions of dollars) Table 7. Table gives the population of a town (in thousands) from 000 to 008. What was the average rate of change of population (a) between 00 and 00, and (b) between 00 and 006? Year Population (thousands) Table Download for free at

14 08 CHAPTER FUNCTIONS For the following eercises, find the average rate of change of each function on the interval specified. 8. f () on [, ] 9. h() on [, ] 0. q() on [, ]. g () on [, ]. _ on [, ]. p(t) (t )(t + ) on [, ] t +. k (t) 6t + on [, ] t TECHNOLOGY For the following eercises, use a graphing utilit to estimate the local etrema of each function and to estimate the intervals on which the function is increasing and decreasing.. f () + 6. h() g(t) t t + 8. k(t) t _ t 9. m() n() EXTENSION. The graph of the function f is shown in Figure 8. Maimum X. Y.88 Figure 8. Let f (). Find a number c such that the average rate of change of the function f on the interval (, c) is REAL-WORLD APPLICATIONS. At the start of a trip, the odometer on a car read,9. At the end of the trip,. hours later, the odometer read,. Assume the scale on the odometer is in miles. What is the average speed the car traveled during this trip? 6. Near the surface of the moon, the distance that an object falls is a function of time. It is given b d(t).6667t, where t is in seconds and d(t) is in feet. If an object is dropped from a certain height, find the average velocit of the object from t to t. Based on the calculator screen shot, the point (.,.8) is which of the following? a. a relative (local) maimum of the function b. the verte of the function c. the absolute maimum of the function d. a zero of the function. Let f(). Find the number b such that the average rate of change of f on the interval (, b) is 0.. A driver of a car stopped at a gas station to fill up his gas tank. He looked at his watch, and the time read eactl :0 p.m. At this time, he started pumping gas into the tank. At eactl :, the tank was full and he noticed that he had pumped 0.7 gallons. What is the average rate of flow of the gasoline into the gas tank? 7. The graph in Figure 9 illustrates the deca of a radioactive substance over t das. Amount (milligrams) A 0 0 Time (das) Figure 9 Use the graph to estimate the average deca rate from t to t. 0 t Download for free at