In my last post I identified two “problems” related to inverses. The first of these is that there may be no string of operations, no algebra or arithmetic, which tells us how to evaluate the inverse function.
For simple functions you can find the inverse function by switching the x and the y and then solving for y. If you can do that, this produces a nice expression for the inverse. Alas usually you cannot do that.
What to do?
What you can do is invent a name and/or a symbol for the new function. So if f(x) = x2, we write f -1(x) = and we think we have solved the problem. We have not. While I can write , what arithmetic can I do to express this number as a decimal? There is an algorithm for this; (you can find it on the internet by searching for “square root algorithm”). You can use a calculator or look in a table as we did back in the “old days.” But that is not the same as performing a series of arithmetic or algebraic operations.
And if does not slow you down, how about sin-1(0.12345)? The only hope in some cases is to try to solve something like x2 = 10 or sin(y) = 0.12345, not much hope there. We are left with using technology of some sort if we need a number (decimal); calculators have buttons for square roots and inverse sines. But sometimes writing or sin-1(0.12345) is good enough.
Making up a new function or symbol to “solve” a problem, even if that function cannot be written as a string of operations is actually fairly common. The sin(x) is defined as the y-coordinate of a point on the unit circle. Except for some special numbers you cannot find y-coordinates that easily. You have seen others already. All the trigonometric functions and their inverses as well as logarithmic functions are of this sort. Mathematics is full of them.
The next post will discuss the other problem: 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?
This is the second of 5 posts on inverses.