Question

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

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

Answer 1

Answer: (B), (C)

The correct options are
B The range is $\left\{0\right\}\cup \{-1\}\cup \left\{1\right\}$

C The domain is $(-\infty ,0)\cup [1,\infty )$


The figure below represents the graph of the greatest integer function.

$\left[x\right]=0\forall x\in [0,1)$.
Thus, the function $f\left(x\right)$ cannot be defined in this interval.
Thus, the domain of function is $\{-\infty ,0\}\cup [1,\infty )$
The range of function $f\left(x\right)$ is $\left\{0\right\}\cup \{-1\}\cup \left\{1\right\}.$