#### Question

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

#### Solution

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

\[\text{ Since f(x) is defined for all values of } x, x \in R . \]

\[\text{ Or dom } (f(x)) = R\]

\[\text{ Since, x - [x] = {x}, which is the fractional part of any real number } x, \]

\[ f(x) = \sqrt{{x}} . . . . . (1)\]

\[\text{ We know that } \]

\[0 \leq {x} < 1\]

\[ \Rightarrow \sqrt{0} \leq \sqrt{{x}} < \sqrt{1}\]

\[ \Rightarrow 0 \leq f(x) < 1 { \text{ from } (1)}\]

\[\text{ Thus, range of f(x) is } [0, 1) . \]

\[\text{ Or dom } (f(x)) = R\]

\[\text{ Since, x - [x] = {x}, which is the fractional part of any real number } x, \]

\[ f(x) = \sqrt{{x}} . . . . . (1)\]

\[\text{ We know that } \]

\[0 \leq {x} < 1\]

\[ \Rightarrow \sqrt{0} \leq \sqrt{{x}} < \sqrt{1}\]

\[ \Rightarrow 0 \leq f(x) < 1 { \text{ from } (1)}\]

\[\text{ Thus, range of f(x) is } [0, 1) . \]

Is there an error in this question or solution?

Solution Write the Domain and Range of F ( X ) = √ X − [ X ] . Concept: Functions.