How could a hypothetical inverse to the absolute value function be represented?

Question

So I know that $\int f'\left(x\right)\ dx$ would be $f\left(x\right)+C$ because for any value of C, $f'\left(x\right)$ would still be the same. Since it has infinite possibilities, we write a "+C" to show that for any value of C the derivative would still hold true, as the pattern for all the solutions of $\int f'\left(x\right)\ dx$ would have the pattern "f(x) plus a constant" A hypothetical inverse to the absolute value function would also have an infinite number of solutions. (Let's write the inverse absolute value of x as ">x<".) For example, >$\sqrt[2]{2}$< could be $\sqrt[2]{2}$, $-\sqrt[2]{2}$, $\sqrt[2]{2}i$, $-\sqrt[2]{2}i$, $1+i$, etc. I know that all of these solutions also follow a pattern of being $\sqrt[2]{2}$ away from $0$, but how would we write this? How would we show that the infinite solutions for this function follow a pattern the same way the antiderivative of f '(x) follows the pattern of "f(x) plus a constant)"?

Answer 1

Comment made into answer.

As you've stated, what you describe as the "inverse absolute value" are all the complex numbers a distance $|x|$ from the origin. So, under your notation, $$\left\rangle x \right \langle =|x| e^{i \theta},$$ for $\theta \in \mathbb R$, since by varying $\theta$ we trace out the circle of radius $|x|$ round the origin.