Math M110: Lecture Notes For Chapter 12 Section 12.1: Inverse and Composite Functions Inverse function (interchange x and y): Find the equation of the inverses for: y = 2x + 5 ; y = x 2 + 4 Function: (Vertical line crosses only once) profit profit year year A B C Graph A: A function, vertical line crosses only once. Different input, Different output. Graph B: A function, vertical line crosses only once. Different input, Same output (the profit in two different years were the same). Graph C: Not a function, vertical line crosses more than once. Same input, Different output (two different profits for the same year). One-to-One Function: (Horizontal line crosses only once, Different input, Different output) A B C Graph A and C: One-to-one function, horizontal line crosses only once. Different input, Different output. Graph B: Not one-to-one function, horizontal line crosses more than once. Different input, Same output. Note: If the function is not one-to-one, then the inverse is not a function. Graph B is not one-to-one and if we graph its inverse, we get graph C which is not a function The following examples will be solved in class from page 792: 42, 48, 52, 54, 56, 58 1
Composite functions The following examples will be solved in class from page 792: 10, 12, 16, 18, 20 Example 1: for f(x) = 3x + 2 and g(x) = 2 + x 2. Find: a) f o g( x ) b) g o f ( x ) c) f o g( 5 ) d) g o f ( 5 ) Example 2: for f(x) = 4x + 3 and g(x) = 2x 2-2. Find: a) f o g( x ) b) g o f ( x ) c) f o g( 1) d) g o f ( 1) Section 12.2 Graphs of exponential Functions f(x) = a x Graph: y = 3 x Graph: y 1 = 4 x The following examples will be solved in class from page 800: 12, 14, 18, 22, 26, 28, 34, 42, 44, 48 2
Section 12.3: Logarithmic Functions 1. Convert the following from exponential form to logarithmic form: a) y= x 2 Answer: log y= 2 x b) 2 3 = 8 Answer: log 2 8= 3 c) 9 1/2 = 3 Answer: log 9 3= 1/2 d) 10 2 = 100 Answer: log 10 100= 2 2. Convert the following from logarithmic to exponential form: a) log 9= 3 x Answer: x3 = 9 b) log 5 25= 2 Answer: 5 2 = 25 c) y = log 2 4 Answer: 2 y = 4 d) log a x= y Answer: a y = x Table 1: log 9 x = 1 x = 9, because 9 1 = 9 or log 9 9 = 1 log 3 x = 1 x = 3, because 3 1 = 3 or log 3 3 = 1 log a x = 1 x = a, because a 1 = a or log a a = 1 log a a = 1 Table 2: log 9 x = 0 x = 1, because 9 0 = 1 or log 9 1 = 0 log 3 x = 0 x = 1, because 3 0 = 1 or log 3 1 = 0 log x x = 0 x = 1, because x 0 = 1 or log x 1 = 1 log a 1 = 0 3. Find: a) log 10 1000 b) log 5 25 c) log 10 1 d) log 10 10 3
4. Graph the following : a) y = log 2 x same as graphing: 2 y = x b) y = log 7 x same as graphing: (7) y = x 5. Solve for x: a) log 2 x = 5 Answer: x = 2 5 =32 b) log 5 x = 0 Answer: x = 1 c) log 2 x = -3 Answer: x = 1/8 d) log 27 x = 1/3 Answer: x = 3 The following examples will be solved in class from page 808: 12, 14, 18, 22, 26, 28, 80, 84, 86, 88, 90 4
Section 12.4: Properties of Logarithmic Functions y = log a x a = x y The number a is called the logarithmic base If a = 10, then it is log 10 and it is called Common logarithm (available in calculator as log) If a = e, then it is log e or ln and it is called Natural logarithm (available in calculator as ln) log 10 x = log x ( Common Logarithm) loge x = ln x ( Natural Logarithm) Properties of logarithms Rule Formula I) Multiplication Rule: log of multiplication = sum of log log(m.n) = log M + log N II) Division Rule: log of division = difference of log log (M / N) = log M - log N III) Power Rule log M k = k. log M Important: I) Example: log 5x = log 5 + log x II) Example: log 5/x = log 5 - log x III) Example: log 5 x = x log 5 Wrong log M. log N log M + log N log M / log N log M - log N (log M) k k.log M Correct log(m.n) = log M + log N log (M / N) = log M - log N log M k = k. log M A. Example: Express in term of logarithms: a) log(x 2 y 2 ) 3 x y b) log 4 z c) log 3 x 2 3 y z 2 5
B. Example: Express as a single logarithm: a) 3 log x 1/2 log y + 3 log z b) 2 log x - 3 log y - 2 log z c) log (x 2-9) - log (x + 3) d) log x 2-2 log x C. Example: Given log b 2 = 1.2 and log b 3 = 2.1. Find a) log b 6 1 b) log b 2 c) log b 2b d) log b 3 e) log b 9 Reminder: log a a = 1 ; log e e = 1 or ln e = 1 log a 1 = 0 ; log e 1= 0 or ln1 = 0 The following examples will be solved in class from page 815: 8, 14, 18, 26, 28, 34, 36, 40, 48. 50, 56, 58, 60 6
Section 12.5: Natural logarithmic Function ln A. Example: Use the calculator to find the following: (round the answer to 4 decimal places) a) ln 30 b) log 98.3 c) ln e 3.06 d) e -2.64 Changing logarithm Base: log M b = log log B. Example: Find the following: (round the answer to 4 decimal places) a) log 5 8 b) log 3 7 c) log 2 0.08 C. Example: Graph and state the domain and the range of: a) f(x) = e -x a a M b b) f(x) = ln x - 1 c) f(x) = ln (x + 1) 7
Section 12.6: Solving Equations A. Solve for x: a) 2 x = 2 3 b) 5 x = 125 c) 8 x = 1/2 d) 4 2x-3 = 64 B. Solve for x: a) log x = -4 b) log 9 x = 1/2 c) ln x = -3 d) 5x = 10 e) 3 x = 10 f) ln x = 4 g) 3x = 5 C. Solve for x: a) log (2x- 1) - log 3 3 (x- 4) = 2 b) log (x - 4) + log 3 3 (x+ 4) = 3 c) log x + log (x - 3) = 1 d) log 2 x + log 2 (x 2) = 3 e) log 4 (x + 6) - log 4 x = 2 f) ln(2t + 1) + ln (2t 1) = 0 g) ln(t - 1) = 3 The following examples will be solved in class from page 830: 54, 58, 60 8
12.7: Applications of Exponential and Logarithmic Functions, Growth, Decline Annual Continuous Growth P = Po ( 1 + k ) = Po a where a > 1 t P = Po. e kt Doubling Time = t ln 2 k Decline, Decay P = Po ( 1 k ) = Po b where 0 < b < 1 P = Po. e Half time = t kt ln 2 k t P 0 : the principal, original amount or population P: the new balance, new population t: the time in years k: the rate, (in decimal form) 1. In 2000, the cost of tuition, books, room, and board at a state university is projected to be as: C(t) = P o (a) t =11,054(l.06) t Note: a = 1.06 because of increase of 6% a) Find the college costs in 2005. b) In what year will the cost be $21,000? c) What is the doubling time of the costs? (Answers: $14,793, 2011, 12) 2. Anchorage Population Growth. In 1998, the population of Anchorage, Alaska, reached 253,750, and the exponential growth rate was 2.9% per year. a) Find the exponential growth function. b) What will the population be in 2010? c) In what year will the population be 600,000? d) What is the doubling time? (Answers: P(t) = 253,750.e 0.029t,359,369, 2028, 23.9) 3. Suppose that P o is invested in a savings account in which interest is compounded continuously at 5.4% per year. a) Suppose that $10,000 is invested. What is the balance after 1 yr? 2 yr? 10 yr? b) When will the investment of $10,000 double itself? (Answers: $10,554.85, $11,140.48, $17,160.07, 12.8) 4. Suppose that P o is invested in a savings account in which interest is compounded continuously. a) Suppose that $100,000 is invested and grows to $164,213.96 in 8 yr. Find the interest rate and then the exponential growth function. b) What is the balance after 1 yr? 3 yr? 12 yr? c) What is the doubling time? (Answers: 6.2%, P(t) = 100,000.e 0.062t, $106,396.23, $120,442.23, $210,433.60, 11.2) 5. The population of a city was 250,000 in 1970 and 200,000 in 1980, find: a) The continuous rate and its function. b) Find the half -life if the rate is continuous b) The annual rate and its function. (Answers: 0.223%, P(t) = 250,000.e.-0.002t ; 31 years, 0.2206%, P(t) = 250,000(0.9779) t ) 9