So I've always been told that for a function to be considered explicit it can only have one specific output for each input or simply pass the vertical line test. While I can accept that on it's face I can't help but wonder about using $\pm$ to rewrite implicit equations like the function for a circle.
$1=x^2+y^2$
as this...
$y=\pm\sqrt{1-x^2}$
To essentially make it into a explicit equation that returns two answers. I have a feeling that $\pm$ is just simply not an operator in the conventional sense but is short hand to avoid unnecessary writing. Does anyone have any hard facts in regards to this?
The difference between an implicit function and an explicit function can best be understood by the english definitions of "implicit" and "explicit". I refer you to the English department for greater details there.
An implicit function is simply a function where the dependency is implied but not expressly stated. For example, $x-y-1=0$ could be such that $y$ is a function of $x$ implicitly. However, if I say $y=f(x)=x-1$, I am explicitly stating that $y$ is a function of $x$. There is not much more to know here. A function can be either explicitly stated or implicitly stated.
Note that for the case of $y=\pm\sqrt{1-x^2}$, certainly $y$ is not a function of $x$.