Write the domain and range of f(x)

Question:

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

Solution:

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

Since $f(x)$ is defined for all values of $\mathrm{x}, x \in R$.

Or $\operatorname{dom}(f(x))=R$

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

$f(x)=\sqrt{\{x\}} \ldots \ldots(1)$

We know that

$0 \leq\{x\}<1$

$\Rightarrow \sqrt{0} \leq \sqrt{\{x\}}<\sqrt{1}$

$\Rightarrow 0 \leq f(x)<1 \quad\{$ from $(1)\}$

Thus, range of $f(x)$ is $[0,1)$.

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