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.

  • $\begingroup$ Wow I didn't think this would be that difficult to answer, I thought it was pretty standard mathematical notation, guess I was wrong. $\endgroup$ – user561159 Aug 11 '18 at 18:58
  • $\begingroup$ I think the difficulty is in stating precisely what one means by "...circumstances where it fails to be true." If you can write out a full paragraph or so stating in unequivocal ways a test clarifying when it is true and when it is not, then perhaps we can help. Otherwise, your request is simply ill-defined and cannot have an answer. $\endgroup$ – David G. Stork Aug 11 '18 at 19:18
  • $\begingroup$ It can very easily have an answer, it's already stated, the only thing left is how it's notated. It seems like you're intentionally being hostile for no reason, so I'll reiterate the facts: At the very least, the statement is true for functions over a domain such that they are defined for all values and invertible. What I have not seen conventionally notated in particular are the branches. The square root function has multiple branches, but despite that, the statement is still true for either one branch or the other at a time, just not both at once since such a map is no longer of a function. $\endgroup$ – user561159 Aug 11 '18 at 19:44
  • $\begingroup$ I don’t see this as a matter of notation but of clarity and precision of language. $\endgroup$ – Lubin Aug 11 '18 at 19:54
  • $\begingroup$ But that's the problem, the question asking how to notate it precisely because there is very specific terminology that matches it. I haven't found myself and I don't specialize in functional analysis and I would have expected the math community here to be competent enough to answer what should be a trivial question, but it seems I was mistaken. $\endgroup$ – user561159 Aug 11 '18 at 20:53

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).


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