Question

Write the domain and range of  \[f\left( x \right) = \sqrt{x - \left[ x \right]}\] .

 

Answer 1

\[f\left( x \right) = \sqrt{x - \left[ x \right]}\]

\[\text{ Since  f(x) is defined for all values of }  x, x \in R . \]
\[\text{ Or dom }  (f(x)) = R\]
\[\text{ Since, x - [x] = {x}, which is the fractional part of any real number } x, \]
\[ f(x) = \sqrt{{x}} . . . . . (1)\]
\[\text{ We know that }  \]
\[0 \leq {x} < 1\]
\[ \Rightarrow \sqrt{0} \leq \sqrt{{x}} < \sqrt{1}\]
\[ \Rightarrow 0 \leq f(x) < 1 { \text{ from } (1)}\]
\[\text{ Thus, range of f(x) is } [0, 1) . \]