There is only one more operation to describe and see how to differentiate where it occurs, and we will be able to differentiate every function we want to differentiate. And that operation is inversion.
We can consider the action of taking an argument and forming the value of f at that argument as an operation on numbers. The inverse of that operation is the act of going backwards and recreating the argument, x, from the value, f(x), of f.
We have encountered this notion before. A typical example of inversion is the square root. The square root function is the inverse of the square function.
Now if you go from the argument to the value and then go back to the argument, you are right back where you started from. And this is the defining property of the inverse function to f. f inverse, applied to f(x) is x again. And it works in the other order as well. If f is inverse to g then g is inverse to f.
This concept has three complications that you must learn to handle. First, is the question of notation. We are tempted to use the notation f-1 for the inverse function to f, and we often do this. But we shouldn't and often we don't use that notation, because it can be confused with the reciprocal function, .
The commonest inverse functions are, the inverses to powers like which are called roots and denoted as , the inverses to the exponent function, exp(x), which is called the natural logarithm of x and denoted as ln(x); the inverse sine function is called the arcsine and is denoted as arcsin(x). On most spreadsheets it is written as =asin(B6), (if you want the arcsine of what is in box B6.) (and there are other related trigonometric inverses which we will talk about when the functions are defined.)
The trouble with using to denote the inverse to f is that this notation is sometimes used for the function, , which is the reciprocal function to f, which you apply to argument x by dividing 1 by f(x), that is by dividing 1 by the value of f. This is totally different from the inverse function to f, which is gotten by switching f 's values and arguments.
The second complication is that the inverse function is not in general defined everywhere. A function like the exponent, exp(x), or the square, whose values are always non-negative, will, upon interchanging values and arguments, only be capable of definition for non-negative arguments. All the other functions we have been considering so far, can be defined almost everywhere; inverse functions, however, often have restricted domains.
The final complication is that many functions that we like to invert take on the same value for more than one argument. The function, f with f(x) = x2, the function that squares, taking x to x2, is a good example of this. 5 and -5 have the same square. Which should be called the square root of 25?
The sine function is periodic and repeats itself endlessly as you go around and around a circle, with period 2. Which of its many arguments which have the same value should be taken as the value of the inverse function to it?
The answer to such questions is that in inverting a function f which takes on the same values more than once, we must first restrict the domain of f so that this does not happen, so that f takes on each value at most once, in its restricted domain. The square function can be restricted to the non-negative numbers, or to the non-positive numbers, (or to appropriate mixtures). After such restriction this problem disappears, since in the restricted domain, f is single valued.
For roots we typically restrict the domain of the power being inverted to be the non-negative numbers, which means that the square root which we call is always positive. In principle we could have chosen to be negative instead, or negative over part of its range and positive on the rest. We do not do this for two reasons: first it is an unnatural thing to do; second, the positive square root has the nice property that the square root of a product, say of xy is the product of their square roots; this is not true for negative square roots, since the product of two of them is positive, and not a negative square root.
In general, what we have been saying means that the inverse function to f requires an added condition to be well defined, when f is not single valued. To get a unique inverse function you must make a restriction of the domain of f to one in which f is single valued.
There are three observations to be made about inverse functions, two nice and the other less nice.
The first nice one is that it is very easy to find the graph of an inverse function from that of the original function, and therefore to decide on a domain for f (which becomes the range for f -1). It is similarly easy to graph f -1 on a spreadsheet.
The way to find the graph of the inverse function is to rotate your paper (which has the graph on it) by degrees around the main diagonal (the line through the origin at angle counterclockwise from the x axis.) You will then find that you have to look through your paper at the function but that can usually be done and if you start with the graph of f you are looking at the graph of the inverse function to f.
For the spreadsheet, you can set up the spreadsheet you use to graph a function, and copy the column of arguments x beyond the column of values f(x), and then highlight and do an x-y scatter chart of the f and new x columns.
8.1 Set up a spreadsheet that plots the exponent function in the domain from -3 to 2. Copy the argument column after the value column for it and highlight the value column and the copied column and plot the inverse function to the exponent, which is the natural log function. For what argument is ln x 0? For what argument is it 1? -1?
8. 2 The applet below allows you to enter functions and plot their inverses as well as themselves. Check your answer to 1 by finding the inverse to exp x in the given domain with the applet.
The not so nice observation is that there is no standard obvious way of finding the value of an inverse function at a particular argument x. All the other functions we have discussed can be found by performing simple standard operations such as adding, dividing, multiplying, subtracting, and substituting. But there is no such procedure for inverses. And in general, there cannot be one. This is because in general you have to choose the domain for the original function to make it single valued, and a means of calculating the inverse would have to know in advance what decision you will make, if it is to get the corresponding inverse.
Of course most inverse functions that you will ever encounter, and perhaps all of them, are accessible as functions on your spreadsheet or calculator. You can compute them by pushing a button. This is because the maker of your machine and its programs has made the decision for you as to what domain to choose for the original function and hence what range to get for the inverse function to it, and has used some sneaky procedure for computing it.