Question

Find the domain of $f\left(x\right)=\sqrt{\left[x\right]-1}+\sqrt{4-\left[x\right]}$
, where [.] denotes GIF

• A. $(1,5)$
• B. $[1,5]$
• C. $[1,5).$

Answer 1

The correct option is C $[1,5).$

Given: $f\left(x\right)=\sqrt{\left[x\right]-1}+\sqrt{4-\left[x\right]}$

From the restriction of square root, $\left[x\right]-1\ge 0$ and $4-\left[x\right]\ge 0$
$\Rightarrow \left[x\right]\ge 1$ and $\left[x\right]\le 4$

From the properties of GIF, $\left[x\right]\ge n\Rightarrow x\ge n\forall n\in Z$

$\therefore \left[x\right]\ge 1\Rightarrow x\ge 1\Rightarrow x\in [1,\infty )\cdots \left(i\right)$

Also, $\left[x\right]\le n\Rightarrow x<n+1\forall n\in Z$

$\therefore \left[x\right]\le 4\Rightarrow x<5$
$\Rightarrow x\in (-\infty ,5)\cdots \left(ii\right)$

Combining $\left(i\right)\&\left(ii\right),$ we get:
$x\in [1,\infty )\cap (-\infty ,5)$
The common portion is $[1,5).$
$\Rightarrow$ Domain is $[1,5).$