8.1 Inverse Functions

The inverse of a function \(f\) is another function \(f_{inv}\) defined so that \(f(f_{inv}(x)) = x\) and \(f_{inv}(f(x)) = x\) both hold.

In words, the inverse function to \(f\) acting on \(f\) produces the identity function, \(x\). Also \(f\) acting on its inverse function is the identity function.

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.

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 is sometimes used to represent the reciprocal function, whose value at argument \(x\) is \(\frac{1}{f(x)}\).

The commonest inverse functions are, the inverses to powers like \(x^k\) which are called roots and denoted as \(x^{\frac{1}{k}}\) and the inverse 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.) There are similarly, \(\arccos(x)\), \(\arctan(x)\), and so on.

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 unless we want to extend our number system.

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 \(f(x) = x^2\), the function that squares, taking \(x\) to \(x^2\), 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 \(2\pi\) (measuring angles in radians.). Which of its many arguments for which sine has the same value should be taken as the value of the inverse function to the sine?

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 this restricted domain, if we want its inverse to be a single valued function. 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 \(f\) is single valued in the restricted domain.

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 write as \(x^{\frac{1}{2}}\), is always positive. In principle we could have chosen \(x^{\frac{1}{2}}\) to be negative instead, or negative over part of its domain 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 is never 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.

One way to find the graph of the inverse function is to rotate your paper (which has the graph on it) by \(\pi\) radians (\(180\) degrees) around the main diagonal (the line through the origin at angle \(\frac{\pi}{4}\) or \(45\) degrees 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 old \(f\) and new \(x\) columns. You will see the graph of the inverse to your function.

Exercises:

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 mathlet 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 mathlet.

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 in doing so, if it is to get the corresponding inverse. This is something it cannot in general do.

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 once it has been determined.