Let $$A=[a, b, c, d], B=[1,2,3]$$. Relation $$R_1, R_2, R_3, R_4$$ are as follows :

$$\begin{aligned} & R_1=[(\mathrm{a}, 1),(\mathrm{b}, 2),(\mathrm{c}, 1),(\mathrm{d}, 2)] \\ & \mathrm{R}_2=[(\mathrm{a}, 1),(\mathrm{b}, 1),(\mathrm{c}, 1),(\mathrm{d}, 1)] \\ & \mathrm{R}_3=[(\mathrm{a}, 2),(\mathrm{b}, 3),(\mathrm{c}, 2),(\mathrm{d}, 2)] \\ & \mathrm{R}_4=[(\mathrm{a}, 1),(\mathrm{b}, 2),(\mathrm{a}, 2),(\mathrm{d}, 3)] \text {, then } \end{aligned}$$

The domain of the function $$f(x)=\frac{1}{\sqrt{x+|x|}}$$ is

The range of the function $f(x)=\frac{x-3}{5-x}, x \neq 5$ is

For $$f(x)=[x]$$, where $$[x]$$ is the greatest integer function, which of the following is true, for every $$x \in \mathbf{R}$$