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Question
Write the domain and range of \[f\left( x \right) = \sqrt{x - \left[ x \right]}\] .
Solution
\[\text{ Or dom } (f(x)) = R\]
\[\text{ Since, x - [x] = {x}, which is the fractional part of any real number } x, \]
\[ f(x) = \sqrt{{x}} . . . . . (1)\]
\[\text{ We know that } \]
\[0 \leq {x} < 1\]
\[ \Rightarrow \sqrt{0} \leq \sqrt{{x}} < \sqrt{1}\]
\[ \Rightarrow 0 \leq f(x) < 1 { \text{ from } (1)}\]
\[\text{ Thus, range of f(x) is } [0, 1) . \]
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