4 1 Composite and inverse functions composite fx

  • Slides: 9
Download presentation
4. 1 Composite and inverse functions composite f(x) = 2 x – 5 g(x)

4. 1 Composite and inverse functions composite f(x) = 2 x – 5 g(x) = x 2 – 3 x + 8 (f◦g)(7) = decomposing h(x) = (2 x -3)5 (f◦g)(7) = f(g(7)) = f(72 – 3∙ 7 + 8) = f(36) = 2∙ 36 – 5 = 67 f(x) = x 5 g(x) = 2 x – 3 (f◦g)(x) = (2 x – 3)5 = h(x) f(x) = ? g(x) = ? inverse relation f(x) = x 2 – 6 x f-1(x) = ? y = x 2 – 6 x (switch x and y) x = y 2 – 6 y x + 9= y 2 – 6 y + 9 (x + 9) = (y - 3)2 (x + 9)½ = y – 3 (x + 9)½ + 3= y = f-1(x) f(x) = 5 x + 8 f-1(x) = (x-8) 5 (f-1◦f)(x) = f-1(f(x)) = = f-1 (5 x + 8) = = (5 x + 8) – 8 5 = 5 x = x 5 (f-1◦f)(x) = ?

4. 2 Exponential Functions and Graphs Compound Interest A is the amount of money

4. 2 Exponential Functions and Graphs Compound Interest A is the amount of money that a principal P will be worth after t years at an interest rate of i, compounded n times a year. A = P(1 + i/n)nt y = 2 x y=x x = 2 y $100, 000 is invested for t years at 8% interest compounded semiannually. e A = $100, 000(1 +. 08/2)2 t A = $100, 000(1. 04) )2 t t= 0 A = $100, 000 t= 4 A $136, 856. 91 t= 8 A $187, 298. 12 t=10 A $219, 112. 31 2. 718284. . .

4. 3 Logarithmic Functions and Graphs For any exponential function f(x) =ax, it inverse

4. 3 Logarithmic Functions and Graphs For any exponential function f(x) =ax, it inverse is called a logarithmic function, base a f(x)=2 x f-1(x) = log 2 x f(x)=2 x f(x) = x f-1(x) = log 2 x Write x = ay as a logarithmic function loga 1 = 0 and logaa = 1 for any logarithmic base a logax = y if the base is 10 then it is called a common log

4. 3 Logarithmic Functions and Graphs (cont) f(x)=ex For any exponential function f(x) =ex,

4. 3 Logarithmic Functions and Graphs (cont) f(x)=ex For any exponential function f(x) =ex, it inverse is called a natural logarithmic function f-1(x) = ln x logb. M = log M The Change of base formula a logab Write x = ey as a logarithmic function loge 1 = ln 1 = 0 and logee = ln e = 1 for any logarithmic base e ln x = y if the base is e then it is called a natural log

4. 4 Properties of Logarithmic Functions Product Rule loga. MN = loga. M +

4. 4 Properties of Logarithmic Functions Product Rule loga. MN = loga. M + loga. N Power Rule loga. Mp = ploga. M The Quotient Rule Logarithm of a Base to a Power loga. M/N = loga. M - loga. N loga ax = x

4. 4 Properties of Logarithmic Functions cont. A Base to a Logarithmic Power alogax

4. 4 Properties of Logarithmic Functions cont. A Base to a Logarithmic Power alogax = x loga 75 + loga 2 loga 150 ln 54 – ln 6 ln 9 5 log 5(4 x-3) 4 x - 3

4. 5 Solving Exponetial and Logarithmic Equations For any a>0, a 1 Base –

4. 5 Solving Exponetial and Logarithmic Equations For any a>0, a 1 Base – Exponent Property ax = a y x=y 23 x-7 = 25 3 x-7 = 5 3 x = 12 x=4 3 x = 20 log 3 x = log 20 x log 3 = log 20 x = log 3 / log 20 x 2. 7268 ex – e-x – 6 = 0 ex + 1 / e x – 6 = 0 e 2 x + 1 – 6 ex = 0 e 2 x – 6 ex + 1 = 0 ex = 3 8 ln ex = ln (3 8) 1. 76

4. 6 Applications and Models: Growth and Decay Exponential growth Population P(t) = P

4. 6 Applications and Models: Growth and Decay Exponential growth Population P(t) = P 0 ekt where k>0 In 1998, the population of India was about 984 million and the exponential rate of growth was 1. 8% per year. What will the population be in 2005? P(7) = 984 e 0. 018(7) P(7) 1116 million Interest Compounded Continuously P(t) = P 0 ekt $2000 is invested at an interest rate k, compounded continuously, and grows to $2983. 65 in 5 years. What is the interest rate? P(5) = 2000 e 5 k $2983. 65 = $2000 e 5 k 1. 491825 = e 5 k ln 1. 491825 = 5 k k 0. 08 or 8%

4. 6 Applications and Models: Growth and Decay cont. k. T = ln 2

4. 6 Applications and Models: Growth and Decay cont. k. T = ln 2 Growth Rate and Doubling Time k = ln 2/T T = ln 2/k Logistic Function Models of Limited Growth P(t) = a. 1 + be-kt Exponential Decay P(t) = P 0 e-kt where k>0 Converting from Base b to Base e bx = e x(lnb)