Question

The real-valued function $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{cosec}}^{-1}\left(\frac{\mathrm{x}}{\sqrt{\mathrm{x}-\left[\mathrm{x}\right]}}\right)$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is defined for all $x$ belonging to:

• A. All non- integers except the interval $[–1,1]$
• B. All integers except $0,–1,1$
• C. All reals except integers
• D. All reals except the interval $[–1,1]$

Answer 1

The correct option is A

All non- integers except the interval $[–1,1]$

Compute the value:

Given , $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{cosec}}^{-1}\left(\frac{\mathrm{x}}{\sqrt{\mathrm{x}-\left[\mathrm{x}\right]}}\right)$

We can see that, $\mathrm{x}\in (-\infty ,-1]\cup [1,\infty )-$all integers

It shows that $\left\{\mathrm{x}\right\}\ne 0$ and $\mathrm{x}\ne$integer because $\mathrm{f}\left(\mathrm{x}\right)$ is undefined for these values.

Hence option (A) is the correct option.