Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 45. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | PART IV Inverse Functions A Case Study of Exponential and Logarithmic Functions C H Inverse Functions Can What Is Done Be Undone What s done cannot be undone. MacBeth Act V scene 1 Now mark me how I will undo myself. Richard II Act IV scene 1 WHAT DOES IT MEAN FOR F AND G TO BE INVERSE FUNCTIONS Some actions can be undone other actions once taken can never be undone. If we think of a function as an action on an input variable to produce an output we can make a similar observation. Let s begin by looking at functions whose actions can be undone. If a function acts on its input by adding 3 the action can be undone by subtracting 3. If a function acts by doubling its input the action can be undone by halving. If f is a function whose action can be undone we refer to the function that undoes the action of f as its inverse function and denote it by f -1. We read f-1 as f inverse. CAUTION This notation while quite standard can be very misleading. f inverse is not - F- The reciprocal of f x is written -tA or f x -1. J x J x 421 422 CHAPTER 12 Inverse Functions Can What Is Done Be Undone EXAMPLE Below are some simple functions together with their inverse functions. Function i. x x 3 ii. g x 2x iii. A x x3 Action done Add 3 to input Double input Cube the input Action undone Subtract 3 from input Halve input Take the cube root Inverse function -1 x x - 3 g-1 x x 2 A-1 x x1 3 Notice again that the inverse of is not the reciprocal1 of . What exactly do we mean when we say the inverse function of undoes the action of If assigns to the input a the output b then its inverse -1 assigns to the input b the output a. That is if a b then -1 b a. Equivalently for every point a b on the graph of the point b a lies on the graph of -1. If and 1 are inverse functions 1 undoes the action of and vice versa so a 4 b h-1 a and b h- a 4 b f a b input of f output of f f 1 b a output of f input of f -1 Figure Look at Example i where x x 3 and 1 x x - 3. 4 f 7 f-1 4 5 4-1 2 4 5
