Question

The domain of ${\cos }^{-1}(x^2-4)$ is

• A. $[3,5]$
• B. $[-1,1]$
• C. $[-\sqrt{5},-\sqrt{3}]\cup [\sqrt{3},\sqrt{5}]$
• D. $[-\sqrt{5},-\sqrt{3}]\cap [-\sqrt{5},\sqrt{3}]$

Answer 1

The correct option is C $[-\sqrt{5},-\sqrt{3}]\cup [\sqrt{3},\sqrt{5}]$

Given, $y={\cos }^{-1}(x^2-4)$

$\Rightarrow \cos y=x^2-4$

i.e.,$-1\le x^2-4\le 1$ $[\because -1\le \cos y\le 1]$

$\Rightarrow -1+4\le x^2\le 1+4$

$\Rightarrow 3\le x^2\le 5$

$\Rightarrow \sqrt{3}\le x\le \sqrt{5}$ and $-\sqrt{5}\le x\le -\sqrt{3}$

By combining the above, we get,

$\Rightarrow x\in [-\sqrt{5},-\sqrt{3}]\cup [\sqrt{3},\sqrt{5}]$