Question

The domain of the function $f\left(x\right)=\sqrt{\frac{4−x^2}{\left[x\right]+2}}$ . where [.r] denotes the greatest integer not more than $x$. is

• A. $(−∞,−2)∪(1,2)$
• B. $(−∞,−2)∪(−1,2)$
• C. $(−∞,−2)∪[−1,2]$
• D. $(−∞,−1)∪(1,2)$

Answer 1

Answer: (C)

Given function $f(x)=\sqrt{\frac{4−x^2}{[x]+2}}$, is define if
$\frac{4−x^2}{[x]+2}â‰\yen 0$
$⇒\frac{x^2−4}{[x]+2}≤0$
So, either $x^2−4≤0$
and $[x]+2>0$ ...(i)
or $x^2−4â‰\yen 0$
and $[x]+2<0$ ...(ii)
From Eq. (i),
$x∈[−2,2]$ and $x∈[−1,∞)$
So, $x∈[−1,2]$ ...(iii)
From Eq. (ii).
$x∈(−∞,−2]∪[−2,−∞]$ and $x∈(−∞,−2)$
So, $x∈(−∞,−2)$ ...(iv)
From intervals Eqs. (iii) and (iv),
$x∈(−∞,−2)∪[−1,2]$