Question

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

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

We have, $f\left(x\right)=\left[\frac{1}{\left[x\right]}\right]$

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