4.1 Composite and inverse functions

1. composite f(x) = 2x – 5 g(x) = x2 – 3x + 8 (f◦g)(7) = (f◦g)(7) = f(g(7)) = f(72 – 3∙7 + 8) = f(36) = 2∙36 – 5 = 67 decomposing h(x) = (2x -3)5 f(x) = ? g(x) = ? f(x) = x5 g(x) = 2x – 3 (f◦g)(x) = (2x – 3)5 = h(x) inverse relation f(x) = x2 – 6x f-1(x) =? y= x2 – 6x (switch x and y) x= y2 – 6y x+ 9= y2 – 6y + 9 (x + 9) = (y - 3)2 (x + 9)½= y – 3 (x + 9)½+ 3= y = f-1(x) f(x) = 5x + 8 f-1(x) =(x-8) 5 (f-1◦f)(x) = ? (f-1◦f)(x) = f-1(f(x)) = = f-1 (5x + 8) = = (5x + 8)– 8 5 = 5x= x 5 4.1 Composite and inverse functions

2. 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 = 2x y = x x = 2y $100,000 is invested for t years at 8% interest compounded semi-annually. A= $100,000(1 + .08/2)2t A= $100,000(1.04) )2t t= 0 A = $100,000 t= 4 A  $136,856.91 t= 8 A  $187,298.12 t=10 A $219,112.31 e 2.7182818284... 4.2 Exponential Functions and Graphs

3. For any exponential functionf(x) =ax, it inverse is called a logarithmic function, base a f(x)=2x f-1(x) = log2x f(x)=2x f(x) = x f-1(x) = log2x Write x = ay as a logarithmic function logax = y loga1 = 0 and logaa = 1 for any logarithmic base a if the base is 10 then it is called a common log 4.3 Logarithmic Functions and Graphs

4. For any exponential functionf(x) =ex, it inverse is called a natural logarithmic function f(x)=ex f-1(x) = ln x logbM = logaM logab The Change of base formula Write x = ey as a logarithmic function ln x = y loge1 = ln1 = 0 and logee = ln e = 1 for any logarithmic base e if the base is e then it is called a natural log 4.3 Logarithmic Functions and Graphs (cont)

5. Product Rule logaMN = logaM + logaN Power Rule logaMp = plogaM The Quotient Rule logaM/N = logaM - logaN Logarithm of a Base to a Power loga ax = x 4.4 Properties of Logarithmic Functions

6. A Base to a Logarithmic Power alogax= x loga75 + loga2 loga150 ln 54 – ln 6 ln 9 5 log5(4x-3) 4x - 3 4.4 Properties of Logarithmic Functions cont.

7. Base – Exponent Property For any a>0, a1 ax= ay x = y 23x-7 = 25 3x-7 = 5 3x = 12 x = 4 3x = 20 log 3x = log 20 x log 3 = log 20 x = log 3 / log 20 x  2.7268 ex – e-x – 6 = 0 ex + 1/ex – 6 = 0 e2x + 1 – 6ex = 0 e2x – 6ex + 1 = 0 ex = 3 8 ln ex = ln (3 8) x = ln (3 8)  1.76 4.5 Solving Exponetial and Logarithmic Equations

8. Exponential growth Population P(t) = P0ekt 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) = 984e0.018(7) P(7)  1116 million Interest Compounded Continuously P(t) = P0ekt $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) = 2000e5k $2983.65 = $2000e5k 1.491825 = e5k ln 1.491825 = 5k k  0.08 or 8% 4.6 Applications and Models: Growth and Decay

9. Growth Rate and Doubling Time kT = ln2 k = ln2/T T = ln2/k Logistic Function Models of Limited Growth P(t) = a . 1 + be-kt Exponential Decay P(t) = P0e-kt where k>0 Converting from Base b to Base e bx = e x(lnb) 4.6 Applications and Models: Growth and Decay cont.