Exponential and Logarithmic. Functions CHAPTER The Algebra of Functions; Composite

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1 CHAPTER 9 Exponential and Logarithmic Functions 9. The Algebra o Functions; Composite Functions 9.2 Inverse Functions 9.3 Exponential Functions 9.4 Exponential Growth and Decay Functions 9.5 Logarithmic Functions 9.6 Properties o Logarithms Integrated Review Functions and Properties o Logarithms A compact luorescent lamp (or light) (CFL) is a type o luorescent light that is quickly gaining popularity or many reasons. Compared to an incandescent bulb, CFLs use less power and last between 8 and 5 times as long. Although a CFL has a higher price, the savings per bulb are substantial (possibly $30 per lie o bulb). Many CFLs are now manuactured to replace an incandescent bulb and can it into existing ixtures. It should be noted that since CFLs are a type o luorescent light, they do contain a small amount o mercury. Although we have no direct applications in this chapter, it should be noted that the light output o a CFL decays exponentially. By the end o their lives, they produce 70 80% o their original output, with the astest losses occurring soon ater the light is irst used. Also, it should be noted that the response o the human eye to light is logarithmic. Electrical Consumption (W) Types o bulbs Electricity Use by Bulb Type Incandescent Compact luorescent Halogen 20V 240V Initial Lumins (lm) 9.7 Common Logarithms, Natural Logarithms, and Change o Base 9.8 Exponential and Logarithmic Equations and Problem Solving In this chapter, we discuss two closely related unctions: exponential and logarithmic unctions. These unctions are vital to applications in economics, inance, engineering, the sciences, education, and other ields. Models o tumor growth and learning curves are two examples o the uses o exponential and logarithmic unctions. (Note: Lower points correspond to lower energy use.) 535

2 536 CHAPTER 9 Exponential and Logarithmic Functions 9. The Algebra o Functions; Composite Functions S Add, Subtract, Multiply, and Divide Functions. 2 Construct Composite Functions. Adding, Subtracting, Multiplying, and Dividing Functions As we have seen in earlier chapters, it is possible to add, subtract, multiply, and divide unctions. Although we have not stated it as such, the sums, dierences, products, and quotients o unctions are themselves unctions. For example, i x2 = 3x and gx2 = x +, their product, x2 # gx2 = 3xx + 2 = 3x 2 + 3x, is a new unction. We can use the notation # g2x2 to denote this new unction. Finding the sum, dierence, product, and quotient o unctions to generate new unctions is called the algebra o unctions. Algebra o Functions Let and g be unctions. New unctions rom and g are deined as ollows. Sum + g2x2 = x2 + gx2 Dierence - g2x2 = x2 - gx2 Product # g2x2 = x2 # gx2 Quotient a g x2 bx2 =, gx2 0 gx2 EXAMPLE I x2 = x - and gx2 = 2x - 3, ind a. + g2x2 b. - g2x2 c. # g2x2 d. a g bx2 Solution Use the algebra o unctions and replace (x) by x - and g(x) by 2x - 3. Then we simpliy. a. + g2x2 = x2 + gx2 = x x - 32 = 3x - 4 b. - g2x2 = x2 - gx2 = x - 2-2x - 32 = x - - 2x + 3 = -x + 2 c. # g2x2 = x2 # gx2 = x - 22x - 32 = 2x 2-5x + 3 d. a x2 bx2 = g gx2 = x - 2x - 3, where x 3 2 I x2 = x + 2 and gx2 = 3x + 5, ind a. + g2x2 b. - g2x2 c. # g2x2 d. a g bx2 There is an interesting but not surprising relationship between the graphs o unctions and the graphs o their sum, dierence, product, and quotient. For example, the graph o + g2x2 can be ound by adding the graph o (x) to the graph o g(x). We add two graphs by adding y-values o corresponding x-values.

3 Section 9. The Algebra o Functions; Composite Functions 537 y ( g)(x) ( g)(3) 7q 0 9 ( g)(6) 0 8 g(6) 6 g(x) 7 ( g)(0) 5 g(3) 5 6 (x) g(0) 4 (6) 4 (3) 2q (0) x Fahrenheit Celsius Celsius 2 Constructing Composite Functions Another way to combine unctions is called unction composition. To understand this new way o combining unctions, study the diagrams below. The right diagram shows an illustration by tables, and the let diagram is the same illustration but by thermometers. In both illustrations, we show Celsius (x) as a unction o Kelvins Fahrenheit x, then Kelvins g(x) as a unction o Celsius x. (The Kelvin scale is a temperature scale devised by Lord Kelvin in 848.) The irst unction we will call, and the second unction we will call g. Table Illustration x Degrees Fahrenheit Input x2 Degrees Celsius Output x Degrees Celsius Input gx2 Kelvins Output x (x) x g(x) g Suppose that we want a unction that shows a direct conversion rom Fahrenheit to Kelvins. In other words, suppose that a unction is needed that shows Kelvins as a unction o Fahrenheit. This can easily be done because the output o the irst unction (x) is the same as the input o the second unction. I we use g((x)) to represent this, then we get the diagrams below. Fahrenheit 3 Celsius x (x) g((x)) g (g ) Kelvins x Degrees Fahrenheit Input g x22 Kelvins Output For example g((- 3)) = 248.5, and so on. Since the output o the irst unction is used as the input o the second unction, we write the new unction as g x22. The new unction is ormed rom the composition o the other two unctions. The mathematical symbol or this composition is g 2x2. Thus, g 2x2 = g x22. It is possible to ind an equation or the composition o the two unctions and g. In other words, we can ind a unction that converts Fahrenheit directly to Kelvins. The unction x2 = 5 x converts Fahrenheit 9 to Celsius, and the unction gx2 = x converts Celsius to Kelvins. Thus, g 2x2 = g x22 = ga 5 9 x b = 5 x

4 538 CHAPTER 9 Exponential and Logarithmic Functions In general, the notation g( (x)) means g composed with and can be written as g 2 (x). Also (g (x)), or g2x2, means composed with g. Composition o Functions The composition o unctions and g is g2x2 = gx22 Helpul Hint g2x2 does not mean the same as # g2x2. c Composition o unctions g2x2 = gx22 while # g2x2 = x2 # gx2 c Multiplication o unctions EXAMPLE 2 I x2 = x 2 and gx2 = x + 3, ind each composition. a. g222 and g 222 b. g2x2 and g 2x2 Solution a. g222 = g222 = 52 = 5 2 = 25 g 222 = g 222 = g42 = = 7 b. g2x2 = gx22 = x + 32 = x = x 2 + 6x + 9 g 2x2 = g x22 = gx 2 2 = x Replace g(2) with 5. [Since gx2 = x + 3, then g22 = = 5.] Since x2 = x 2, then 22 = 2 2 = 4. Replace g(x) with x + 3. x + 32 = x Square x Replace (x) with x 2. gx 2 2 = x I x2 = x 2 + and gx2 = 3x - 5, ind a. g242 b. g2x2 g 242 g 2x2 Helpul Hint In Examples 2 and 3, notice that g 2x2 g2x2. In general, g 2x2 may or may not equal g2x2. EXAMPLE 3 I x2 = 0 x 0 and gx2 = x - 2, ind each composition. a. g2x2 b. g 2x2 Solution a. g2x2 = gx22 = x - 22 = 0 x b. g 2x2 = g x22 = g 0 x 0 2 = 0 x I x2 = x and gx2 = x + 3, ind each composition. a. g2x2 b. g 2x2

5 Section 9. The Algebra o Functions; Composite Functions 539 EXAMPLE 4 I x2 = 5x, gx2 = x - 2, and hx2 = 2x, write each unction as a composition using two o the given unctions. a. Fx2 = 2x - 2 b. Gx2 = 5x - 2 Solution a. Notice the order in which the unction F operates on an input value x. First, 2 is subtracted rom x. This is the unction gx2 = x - 2. Then the square root o that result is taken. The square root unction is hx2 = 2x. This means that F = h g. To check, we ind h g. Fx2 = h g2x2 = hgx22 = hx - 22 = 2x - 2 b. Notice the order in which the unction G operates on an input value x. First, x is multiplied by 5, and then 2 is subtracted rom the result. This means that G = g. To check, we ind g. Gx2 = g 2x2 = g x22 = g5x2 = 5x I x2 = 3x, gx2 = x - 4, and hx2 = 0 x 0, write each unction as a composition using two o the given unctions. a. Fx2 = 0 x b. Gx2 = 3x - 4 Graphing Calculator Explorations Y 3 ax 2 qx 6 Y2 ax Y qx 2 0 I x2 = 2 x + 2 and gx2 = 3 x2 + 4, then + g2x2 = x2 + gx2 = a 2 x + 2b + a 3 x2 + 4b = 3 x2 + 2 x + 6. To visualize this addition o unctions with a graphing calculator, graph Y = 2 x + 2, Y 2 = 3 x2 + 4, Y 3 = 3 x2 + 2 x + 6 Use a TABLE eature to veriy that or a given x value, Y + Y 2 = Y 3. For example, veriy that when x = 0, Y = 2, Y 2 = 4, and Y 3 = = 6. Vocabulary, Readiness & Video Check Match each unction with its deinition.. g2x2 4. g 2x2 2. # g2x2 5. a g bx g2x g2x2 A. g( (x)) B. x2 + gx2 C. (g (x)) D. x2, gx2 0 gx2 E. x2 # gx2 F. x2 - gx2

6 540 CHAPTER 9 Exponential and Logarithmic Functions Martin-Gay Interactive Videos Watch the section lecture video and answer the ollowing questions From Example and the lecture beore, we know that + g2x2 = x2 + gx2. Use this act to explain two ways you can ind + g From Example 3, given two unctions (x) and g(x), can (g(x)) ever equal g( (x))? See Video Exercise Set For the unctions and g, ind a. + g2x2, b. - g2x2, c. # g2x2, and d. a bx2. See Example. g. x2 = x - 7, gx2 = 2x + 2. x2 = x + 4, gx2 = 5x x2 = x 2 +, gx2 = 5x 4. x2 = x 2-2, gx2 = 3x 5. x2 = 2 3 x, gx2 = x x2 = 2 3 x, gx2 = x x2 = -3x, gx2 = 5x 2 8. x2 = 4x 3, gx2 = -6x I x2 = x 2-6x + 2, gx2 = -2x, and hx2 = 2x, ind each composition. See Example g h g h22 3. g h h g202 Find g2x2 and g 2x2. See Examples 2 and x2 = x 2 +, gx2 = 5x 6. x2 = x - 3, gx2 = x 2 7. x2 = 2x - 3, gx2 = x x2 = x + 0, gx2 = 3x + 9. x2 = x 3 + x - 2, gx2 = -2x 20. x2 = -4x, gx2 = x 3 + x x2 = 0 x 0 ; gx2 = 0x x2 = 0 x 0 ; gx2 = 4x x2 = 2x, gx2 = -5x x2 = 7x -, gx2 = 2 3 x I x2 = 3x, gx2 = 2x, and hx2 = x 2 + 2, write each unction as a composition using two o the given unctions. See Example Hx2 = 2x Gx2 = 23x 27. F x2 = 9x Hx2 = 3x Gx2 = 32x 30. F x2 = x + 2 Find (x) and g(x) so that the given unction hx2 = g2x2. 3. hx2 = x hx2 = 0 x hx2 = 2x hx2 = 3x hx2 = 36. hx2 = 2x - 3 x + 0

7 Section 9.2 Inverse Functions 54 REVIEW AND PREVIEW Solve each equation or y. See Section x = y x = y x = 3y 40. x = -6y 4. x = -2y x = 4y + 7 CONCEPT EXTENSIONS Given that -2 = 4 g -2 = = 5 g02 = = 7 g22 = - 72 = g72 = 4 ind each unction value g g g g # g # g a g b a g b-2 5. I you are given (x) and g (x), explain in your own words how to ind g2x2 and then how to ind g 2x Given (x) and g (x), describe in your own words the dierence between g2x2 and # g2x2. Solve. 53. Business people are concerned with cost unctions, revenue unctions, and proit unctions. Recall that the proit P(x) obtained rom x units o a product is equal to the revenue R(x) rom selling the x units minus the cost C(x) o manuacturing the x units. Write an equation expressing this relationship among C(x), R(x), and P(x). 54. Suppose the revenue R(x) or x units o a product can be described by Rx2 = 25x, and the cost C(x) can be described by Cx2 = 50 + x 2 + 4x. Find the proit P(x) or x units. (See Exercise 53.) 9.2 Inverse Functions S Determine Whether a Function Is a One-to-One Function. 2 Use the Horizontal Line Test to Decide Whether a Function Is a One-to-One Function. 3 Find the Inverse o a Function. 4 Find the Equation o the Inverse o a Function. 5 Graph Functions and Their Inverses. 6 Determine Whether Two Functions Are Inverses o Each Other. Determining Whether a Function Is One-To-One In the next three sections, we begin a study o two new unctions: exponential and logarithmic unctions. As we learn more about these unctions, we will discover that they share a special relation to each other: They are inverses o each other. Beore we study these unctions, we need to learn about inverses. We begin by deining one-to-one unctions. Study the ollowing table. Degrees Fahrenheit (Input) Degrees Celsius (Output) Recall that since each Fahrenheit (input) corresponds to exactly one Celsius (output), this pairing o inputs and outputs does describe a unction. Also notice that each output corresponds to exactly one input. This type o unction is given a special name a one-to-one unction. Does the set = 50, 2, 2, 22, -3, 52, 7, 626 describe a one-to-one unction? It is a unction since each x-value corresponds to a unique y-value. For this particular unction, each y-value also corresponds to a unique x-value. Thus, this unction is also a one-to-one unction. One-to-One Function For a one-to-one unction, each x-value (input) corresponds to only one y-value (output), and each y-value (output) corresponds to only one x-value (input).