Mathematics - Functions Question with Solution | TestHub

Given,

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

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

So, $\left[x\right]-x=-\left\{x\right\}$

Now putting the value in given function we get,

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

$\Rightarrow f\left(x\right)=\frac{1}{\sqrt{-\left\{x\right\}}}$

Now we know that $\frac{1}{\sqrt{g\left(x\right)}}$ 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$=\varphi$