The next few posts will concern functions and their inverses. Today we will just get into the basics which, hopefully, students know from their work prior to calculus. Of course, they will have forgotten some of this and claim they never learned it. (Which may be correct, but one hopes they were taught it.) This is one of the places where a brief review just before teaching the calculus of inverses might be useful.
When starting out with functions we are given a value of the input or independent variable, x, and are asked to compute the output value or the dependent variable, y. Then, to mix things up, you may be given an output value and asked to find the input value that produces it. With the beginning functions this is not too difficult; that is, there are algebraic techniques that can be used. If you have to do a number of these, you can easily write an expression solved for x that you can use to find the y’s.
To streamline all this mathematicians use the concept of the inverse of a function. If a function is a set of ordered pairs (a, b) or (x, f (x)) then the inverse is defined as the set with these ordered pairs with the coordinates reversed; (b, a) or (f (x), x). Since this last looks a bit strange we define a new notation f -1(x) to denote the inverse of f(x). That is f -1(x) is the inverse function of the original function f(x). The ordered pairs are now (x, f -1(x)).
For very simple functions the inverse can be found by switching the x and y in the original and the solving for y. So if f (x) = 2x + 3 we write x = 2y + 3 and solve to get y = ½(x – 3). So f -1(x) = (½)(x– 3).
Notice that f (f -1(x)) = 2(½(x – 3))+3 = x.
Several things to note at this point:
- Inverses occur in pairs. The inverse of the inverse is the original function.
- The inverse undoes whatever the function did and returns the original input value. In symbols this is f -1(f (x)) = f (f -1(x))=x.
- The points (a, b) and (b, a) are symmetric to the line y = x and therefore the inverse of a function has a graph that is the reflection of the function’s graph across the line y = x. The function and its inverse are symmetric to this line.
- The domain and range switch places: the domain of the inverse is the range of the function, and the range of the inverse is the domain of the function.
There are also several problems that need to be addressed.
- There may be no string of operations, no “algebra”, which will produce the output for the inverse function. We cannot write an algebraic expression to find the number whose seventh power is 10. What do we do in this case?
- The inverse of a function may not be a function. Since there are two numbers whose square is 9, what is the “unsquare” of 9; is it 3 or –3? When does this happen? Since it is really useful for the inverse to be a function, what can be done about this?
These two “problems” will be the subjects of my next two post.
(This started out as one post and has grown to a series 5.)