JEE 2023 — Mathematics PYQ

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{|x|}-x}$, where $[x]$ denotes the smallest integer greater than or equal to $x$. Then among the statements:

$(S1):A\cap B=(1,\infty )-\mathbb{N}$ and $(S2):A\cup B=(1,\infty )$