… is the square root function.
Why? It’s a perfect example of a function that many people think is well defined, but in reality is not.
What do I mean by “well defined”? I mean a function for which any legal input will provide one, and only one, answer.
The problem is that the square root is considered to be the inverse of squaring something. At best, you can say that the square is the inverse of the square root. This is a subtle, but important, difference.
So: and and . This implies that and and .
This is all fine and dandy — until you consider the negative numbers. Take . Now, if the inverse of the square is the square root, then what is the square root of 81? We see that , so therefore both are candidates for the result of the square root function.
In algebra (where square roots come up often) there are some ways of denoting this. For example, the “number” is taken to mean “either positive or negative 9”. So then the “number” when squared is 81. So you could say that the square root of 81 is . This is still not a real solution to this problem, as all you’ve done is state your uncertainty in a shorthand format. Worse, you have something that looks like a regular equation (), but in reality, you are stating that you have 2 equations, one of which is actually true! (Either or , but not both.) Worse yet, for each in an equation, you have twice as many equations that may (or may not) be true.
There are ways to fix this. One solution is to say that the square root only refers to the positive number, that is, we ignore the negative numbers in taking the square root. This means that the square root is no longer a proper inverse of the square function, since .
Another solution is to add some notation to indicate which square root we are talking about. This is typically denoted by putting a plus or minus over the root symbol, so that and . Again, this means that the square root is no longer a proper inverse of the square.
And this isn’t even considering the square roots of negative numbers… (more on this later)