4 1 Inverse Functions 1 Determine if each pair of functions is an inverse function a 2 5 2 10 b 2 1 1 c 3 2 3 3 3 1 2 Given that what is the domain of 1 1 1 4 5 3 Find for original functions Identify the domain and range of both the inverse and 4 Given each function show your work to find the inverse Identify the domain and range of the function and its inverse Then graph both Clearly label both functions on the graph a 2 3 2 1 b x 1 c 4 4 3 Exponential Logarithmic Functions Graphs 1 Use transformations to make accurate graphs of each function Find the domain and range of each function Find the equation of the asymptote of each function a 2 b c 3 2 d 3 2 e 2 2 f 1 4 4 4 Logarithmic Functions Properties 1 Write in exponential form a log416 2 2 Use change of base to express the logarithm as a quotient of natural logarithms then evaluate to 3 decimal places a log24 b log5625 4 c ln1 0 d log 00001 5 b log5125 c log 100 d ln e 3 Using properties of logarithms expand each expression as much as possible log 2 4 log 100 5 5 3 a b b log x2 y2 2log x y b 2log39 log327 4 Using properties of logarithms rewrite each expression as a single logarithm a 2log x 3log y 5 Using properties of logarithms and change of base evaluate each expression a log24 log216 4 5 Exponential Logarithmic Functions Equations 1 Solve by relating the bases a 4x 16 b 812x 9x 5 c e5x 2 13 2 Solve for x Give both an exact answer and a 3 decimal place approximation when applicable a log 2 x 3 4 2 2 b ln 5x c ln x ln x 3 28 d 2 3x 2 13 e log x 2 log x 1 4 f 32x 5 3x 3 3 g e4x 2 e2x 1 4 6 4 2 Exponential Logarithmic Functions Applications 1 After I left Topeka the population has been decreasing by 8 each year The population of Topeka was 130 000 in 2023 a Write an equation to model the population P as a function of time t b What is the population going to be in 2027 2 You deposit 2 000 into a savings account The annual interest rate is 5 5 Which investment yields the greater return over 5 years the interest rate compounded quarterly or the interest rate compounded continuously How much more is yielded by the better investment to the nearest cent 3 In 2023 you deposit 5 000 into a savings account with an annual interest rate of 4 compounded quarterly At what point in time will your account balance reach 7 000 Round your answer to the nearest tenth of a year 4 You inherit 10 000 and invest it in a mutual fund with an annual interest rate of 8 compounded continuously How many years will it take for your investment to reach 20 000 Round your answer to the nearest tenth of a year 5 A science experiment studies a bacterial culture The bacteria grow exponentially and scientists measure the instantaneous growth rate to be 5 per hour If you start with 1 000 bacteria how long will it take for the population to reach 5 000 bacteria Round your answer to the nearest tenth of an hour 6 An antique camera lens is coated with a special material that gradually degrades over time Scientists believe this degradation follows an exponential decay model with a half life of 50 years If a recently discovered camera lens has only 20 of its original coating remaining how long ago was the camera lens manufactured 7 A freshly baked pie is taken out of the oven at 200 F and placed on a cooling rack in a room that maintains a constant temperature of 70 F After 5 minutes the pie s internal temperature has decreased to 160 F a Find an exact expression for the rate of cooling k of the pie b How long will it take for the pie to cool to a safe handling temperature of 100 F Round your answer to the nearest minute