Question

The domain of the function $f\left(x\right)=\frac{\sqrt{5-x}}{\left(x+1\right)\left(x-2\right)\left(x-3\right)}+a{\sin }^{-1}\left[\frac{2x-5}{3}\right];a\ne 0$ and $[\cdot ]$ denotes greatest integer function, is:

• A. $[1,5]$$-\left\{2,3\right\}$
• B. $\left[1,2\right)\cup \left(3,5\right]$
• C. $\left[1,\frac{11}{2}\right)$
• D. $\left[1,5\right]$

Answer 1

The correct option is A $[1,5]$$-\left\{2,3\right\}$

For ${\sin }^{-1}$ to be defined

$-1\le \left[\frac{2x-5}{3}\right]\le 1$

so, $-1\le \frac{2x-5}{3}<2$

$⟹-1\le 2x-5<6$

$⟹2\le 2x<11$

$⟹1\le x<\frac{11}{2}$

and also, $5-x\ge 0$

$⟹x\le 5$

But $x\ne -1,2,3$

So, the domain will be$f\left(x\right)=[1,5)-\{2,3\}$