Section 1 6 One to one functions Inverse functions and Logarithms Recall the definition of a function To each x there is assigned exactly one y A function is called one to one if Exactly one x is assigned to each y It must be a function first Examples f x mx b if m 0 Example One function which is not one to one is x values g x y x 2 h x x x 3 are all one to one Each non zero y value is assigned to two When a function f is one to one we can always uniquely answer If f a b and we know b what is a because b was assigned to exactly one a For example if f x 2x and f x 7 then what is x x 7 2 If g x 4x 5 and g x 19 then x 6 For the examples f x y mx b x y b m g x y h x y x x 3 x y 2 x 3 y Inverse functions In the above examples if we switch places with x and y we have the inverse functions Inverse function f x y mx b x y b y x b m g x y h x y x x 3 x y m 2 y x 2 y 3 x x 3 y The notation for the inverse function to f x is f 1 x This is similar to the notation for multiplicative inverses but should not be confused Ex If f x 2x then the inverse function is This is not the same as f x 1 2 x 1 f 1 x 1 2 1 2x x 2 1 x When the exponent 1 is on a number it means the multiplicative inverse When it is on the name of the function such as f it means the inverse function Ex h x x 3 h 2 8 h 1 8 2 An important fact to remember f h 8 1 3 1 8 512 1 1 512 f 1 x x and f 1 f x x Ex For h h h 1 1 h x x 3 h 1 1 3 x 3 x x 3 x 3 x x h x 3 The graph of x 3 f x 1 x is a reflection across the line y x of the graph of f You can see this in your calculator by graphing Y1 f x Y2 x and Y3 for each of the examples above The domain of f is the range of f 1 and the rangbe of f is the domain of f f 1 1 x Logarithms For any positive b f x b x is a one to one function So it has an inverse function and the inverse is called log b x the logarithm base b of x When b 10 we write log x Whenever b is not given in the logarithm assume b 10 When b e we write lnx lnx is called the natural logarithm The logarithm is the exponent we raise b to so that power of b will be x The domain of log b x is 0 That is x must be positive Examples log100 2 because 10 2 100 The base is not given so it is 10 and the 2nd power of 10 is 100 log 2 4 2 because 2 log 2 8 3 log 2 16 4 2 2 4 2 2 4 log 3 9 2 8 log 3 27 3 3 27 16 log 3 81 4 3 3 because 3 9 3 4 81 Since exponential grow rapidly for b 1 sometimes logarithmic scales are used Example The richter scale which measure earthquakes is log base 10 scale If the value on the richter scale increases by one then the magnitude of the earthquake is 10 times as great An earthquake which measures 6 on the richter scale is 100 times as strong as one that measures 4 The pH of a solution is log H If the pH increases by 1 then H where H is the hydrogen ion concentration is 1 10th as much Laws of Logarithms These follow from exponent laws log b MN log b M log b N Note There is no rule for log b M log b t M N log b M log b N t t log b M log b M Use parentheses where needed to avoid confusion The logarithm of an exponential function is linear x ln ab ln a x ln b Use logarithms when you need to solve for an exponent Example If any amount of money is invested at 7 annual interest compounded continuously how long will it take to double the original amount To double this we solve Pe 07 t 2 P so e 07 t 2 Since the unknown is in the exponent we take ln of both sided to get A t Pe ln e 07 t 07 t ln 2 07 t ln 2 t ln 2 10 years 07 In general the doubling time for continuous compound interest is ln 2 r where r is the interest rate as a decimal Example Write as a single logarithm and evaluate without a calculator 3 3 log 2 48 log 2 6 First bring down the exponent of 3 from the 6 Then 3 log 2 48 3 log 2 6 3 log 2 48 log 2 6 3 log 2 Solve for x a log 2 x 2 log 2 x 1 log 2 x 1 3 2 1 3 6 3 log 2 8 9 1 3 Solution 3 is the exponent of 2 so that b 48 x 2 1 2 3 8 x 2 9 x 3 or x 3 Using the sum of logs rule we have So we know x 3 or x 3 but now x 3 cannot be substituted in the original expressions so x 3 is the only solution log 2 x