Here is system of equations defined from one domain~codmain but presented onto another (in this case, a proper subset, the reals out of all complex numbers). Is it notated sensically and properly? The inequalities involving (moduli and absvals of) y and x should only restrict (if cause any alteration to) the range of outputs; I am wondering cases when it can add values (discrete, ranges, or both), but more importantly how fully to solve this problem (which I concocted out of self interest, and am okay with modifying per recommendation prior to an answer).
$∀[x],[y]∈{\mathbb{C}}↣ ∃[x∈{\mathbb{R}}]<y, ∀(x,y)∈{\mathbb{R}^2}:$
$|y|<|2x|$
$y=$+$\sqrt{2x+1}$
$x={(y-1)}^4-1$