How do you notate the circumstances of functions relevant to where they are invertible and defined?

Question

I want to state a particular claim such as

"functions in the form $f(x)=ag(x)^2-b$ have a solution in the form of $x=g^{-1}( \pm \frac{ \sqrt{f(x)+b}}{ \sqrt{a}})$

People know that not literally all functions are always defined and invertible over all real numbers to satisfy that solution, but it should be beyond exceptionally obvious that this claim would never be referring to a function in a circumstance where it fails to be true.

How do I notate that the statement is always true for anywhere a function is defined and that it is only using a single branch in the event that such a function would otherwise fail to be invertible? I'm not even talking about any analytic continuation or calculus so the functions $f(x)$ and $g(x)$ wouldn't even need to be continuous in this circumstance.

Answer 1

There does not exist any standard notation for this. It's just not something that comes up much. You instead have to just describe in words what you mean, as you have done.

The way I would recommend writing the claim you seem to be making is something like the following.

If $f$ is a function of the form $f(x)=ag(x)^2+b$, then we can solve for $x$ in terms of $y=f(x)$ by taking any $x$ such that $g(x)=\pm \frac{ \sqrt{y+b}}{ \sqrt{a}}$.

This avoids the problematic use of $g^{-1}$ when you are not actually making any claim about the invertibility of $g$. It also makes it a lot clearer what you mean by "solving a function" (though that may be clear from context in what you are writing).