Question

The domain of ${\cos }^{-1}[2x^2-3],$ where $[.]$ denotes the greatest integer function, is

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

Answer 1

The correct option is C $(-\sqrt{\frac{5}{2}},-1]\cup [1,\sqrt{\frac{5}{2}})$

For ${\cos }^{-1}$ to be defined,
$-1\le [2x^2-3]\le 1$
$\Rightarrow -1\le 2x^2-3<2$
$\Rightarrow 1\le x^2<\frac{5}{2}$
$\Rightarrow x\in (-\sqrt{\frac{5}{2}},-1]\cup [1,\sqrt{\frac{5}{2}})$