Math 103 Benchmark 2 Review Solutions 1 Chapter 4 1 De ne what it means for a function to have an inverse What does this mean in terms of inputs and outputs How can we determine if a function has an inverse by looking at its graph Solution Suppose f x has an inverse function denoted f 1 x Then f x y if and only if f 1 y x In terms of inputs and outputs the inputs f 1 x are the outputs of f x and the outputs of f 1 x are the inputs of f x We can determine if a function has an inverse by seeing if its graph passes the horizontal line test 2 Find the inverse of the following functions If the inverse does not exist explain why i f x 5 x 2 Solution ii g t t 7 2 iii h y y 8 3 17 Solution 5 x 2 y f x y x 2 5 x 2 5 y 5 y 5 y x f 1 y 2 2 y 8 3 x g t 17 3 17x y 8 3 17x y 8 3 17x 8 y f 1 x 3 17x 8 1 Solution This function is a quadratic function which has a parabola as a graph A parabola does not pass the horizontal line test so this function does not have an inverse Math 103 Benchmark 2 Review Solutions iv m x 4e5x Solution y m x 4e5x y 4 cid 17 cid 17 cid 1 cid 16 y cid 16 y ln cid 0 y 4 4 4 ln ln 5 f 1 y e5x ln e5x 5x x cid 1 ln cid 0 y 4 5 3 Fill in the properties of logarithms cid 16 x cid 17 a logb xy logb x logb y logb x logb y b logb c logb xk k logb x y d logb b 1 e blogb x x 2 Math 103 Benchmark 2 Review Solutions 4 Simplify the following logarithms using logarithm properties a log 4 log 22 2 log 2 b log3 6 log3 2 3 log3 2 log3 3 log3 2 1 c logb 3b b2 logb b 3 b logb b logb b 3 1 logb 3 b cid 17 cid 16 x2 3y3 d log log x2 log 3y3 2 log x cid 0 log 3 log y3 cid 1 2 log x log 3 3 log y 3 Math 103 Benchmark 2 Review Solutions 5 Evaluate the following without a calculator a log cid 0 1 cid 1 log 10 4 4 log 10 4 104 b log3 81 log3 34 4 log3 3 4 cid 17 cid 16 e7 3e2 cid 17 cid 16 e5 3 c ln ln ln e7 ln 3 7 ln 3 d log 10000000000 log 1010 10 log 10 10 6 Solve the equations a 18 2 log3 x Solution b e4t 7 Solution 18 2 log3 x 9 log3 x 39 3log3 x 39 x e4t 7 ln e4t ln 7 4t ln 7 ln 7 t 4 4 Math 103 Benchmark 2 Review Solutions c 100 log 3x Solution d 256 ln e x 3 Solution 100 log 3x 10100 10log 3x 10100 3x 10100 x 3 256 ln e x 3 x 256 3 3 256 x 768 x 5 Math 103 Benchmark 2 Review Solutions 7 Let P t 800 75 t be the amount of bacteria left in an experiment after t hours What is the half life for the bacteria Be sure to include units Solution Since we are solving for half life we are solving for the time it takes for the amount of bacteria to reach half its initial amount Then we have So the half life of the bacteria is years ln 0 5 ln 0 75 8 Suppose the population of bacteria is given by B t 40e3t after t hours What is the doubling time Be sure to include units Solution Since we are nding the doubling time we are nding the time it takes for the population of bacteria to reach double its initial amount Then we have So the doubling time is hours ln 2 3 9 If a population doubles in size every 5 years nd the percent annual growth of the population Then nd the continuous percent growth rate of the population Solution To nd an equation for the percent annual growth we use P t abt Since the doubling time is 5 we know that when t 5 the population will be 2a Then we have Note that this is the growth factor To nd the growth rate we take b 1 5 2 1 To nd the continuous growth rate we would need the growth rate from the equation P t P ert But since we know that b 5 2 Then to solve we take the natural logarithm of both sides to get r ln 5 2 2 we know that er b 5 400 800 0 75 t 0 5 0 75 t ln 0 5 ln 0 75t ln 0 5 t ln 0 75 ln 0 5 ln 0 75 t 80 40e3t 2 e3t ln 2 ln e3t ln 2 3t ln 2 t 3 P t abt 2a P 5 ab5 2 b5 5 2 b 6 Math 103 Benchmark 2 Review Solutions 2 Chapter 5 10 List the possible transformations that can be applied to the graph of f x Be as explicit as possible for example what the equation would look like depending on whether something is positive or negative Solution We have covered the following transformations 1 Re ections f x is a re ection across the x axis and f x is a re ection across the y axis 2 Vertical shifts f x k is a vertical shift up by k units if k is positive but it is a shift down by k units if k is negative 3 Horizontal shifts f x h is a horizontal shift to the right by h units if h is positive but it is a shift to the left by h units if h is negative 4 Vertical stretches compressions af x is a vertical stretch by a factor of a if a 1 but it is a if a 1 In these cases a is positive since if it were a vertical compression by a factor of 1 negative that would fall under re ections 5 Horizontal stretches compressions f bx is a horizontal stretch by a factor of 1 b if b 1 but it is a horizontal compression by a factor of b if b 1 In these cases b is positive since if it were negative that would fall under re ections 11 Give the order of transformations for f x if g x a f b x c d where a b c and d are positive constants Solution When deciding the order we rst pick either vertical or horizontal transformations Once we choose we nish that set Then we move on to …