Question

The function $f\left(x\right)=\frac{{\sec }^{-1}x}{\sqrt{x-\left[x\right]}}$, where $\left[x\right]$ denotes greatest integer function less than or equal to $x$ is defined for $x$ belongs to

• A. $R$
• B. $R-\{(-1,1)\cup n:n\in I\}$
• C. $R^{+}-(0,1)$
• D. $R^{+}-(n:n\in N)$

Answer 1

Answer: (B)

$R-\{(-1,1)\cup n:n\in I\}$

Note that ${\sec }^{-1}x$ is defined for $x\in (-\infty ,-1]\cup [1,\infty )$.

Also the function is not defined when $\sqrt{x-\left[x\right]}=0$.

Hence the function is not defined when $x=\left[x\right]$ or when $x\in I$

$\therefore$ The function is defined for $xwherex\in R-\{(-1,1)\cup n:n\in I\}$