Question

Find the domain of the real valued function: f(x) = $\sqrt{x - [x]}$

Answer 1

The $\textbf{domain}$ of function $\textbf{f(x)}$ is the set of all values for which function is defined.

we have to find domain of real valued function : f(x) = $\sqrt{x-[x]}$

here, [x] denotes greatest integer function or floor function of x.
we know, greatest integer function or floor function is defined to be the greatest integer less than or equal to the real number x.

and x - [x] denotes fractional part of x.
fractional part of x is nonnegative function.
it lies between 0 to 1

e.g., 0 ≤ x - [x] ≤ 1 for all real value of x ......(1)

To define f(x),

x - [x] ≥ 0 ......(2)

from equation (1) and (2),

we get, x $\in (-\infty,\infty)$

hence, domain of function $\in (-\infty,\infty)$