Question

For the function $f\left(x\right)=\left[\frac{1}{\left[x\right]}\right]$ , where $[x]$ denotes the greatest integer less than or equal to $x$, which of the following statements are true ?

• A. The domain is $\left(-\infty ,\infty \right)$
• B. The range is $\left\{0\right\}\cup \left\{-1\right\}\cup \left\{1\right\}$
• C. The domain is $\left\{-\infty ,0\right\}\cup [1,\infty )$
• D. The range is $\left\{0\right\}\cup \left\{1\right\}$

Answer 1

We have, $f(x)=\left[\frac{1}{[x]}\right]$
Domain $=R-\{f(x)=0\}$
Now, $f(x)$ will be zero, when $[x]=0$
$\Rightarrow x\in [0,1)$
$\therefore$ Domain of $f(x)=R-(0,1]$ and range of $f(x)=\{-1,0,1\}$