Let the sets A and B denote the domain and range respectively of...

Given,

$f\left(x\right) = \dfrac{1}{\sqrt{\left[x\right] - x}}$

We know that, $x = \left[x\right] + \left\{x\right\}$ where \left\{x\right\}$is fractional part of$x$whose value is$0 < \left\{x\right\} < 1$,</p><p>So, \left[x\right] - x = - \left\{x\right\}$

Now putting the value in given function we get,

$f$\left(x\right) = \dfrac{1}{\sqrt{\left[x\right] - x}}

$⇒ f\left(x\right) = \dfrac{1}{\sqrt{- \left\{x\right\}}}$

Now we know that $\dfrac{1}{\sqrt{g\left(x\right)}}$ is defined when $g\left(x\right) > 0$, but $- \left\{x\right\} \in \left(- 1, 0\right)$

Hence, $\sqrt{- \left\{x\right\}}$ is not defined

Hence, Domain and Range will be$= ϕ$