Question

If $f\left(x\right)={\sin }^{-1}\sqrt{x-4}$, then the range of $x$ is

• A. $\left[4,5\right]$
• B. $\left[-5,-4\right]$
• C. $\left[-1,1\right]$
• D. $\left[0,1\right]$

Answer 1

The correct option is A

$\left[4,5\right]$

Explanation for the correct option.

Find the range of $x$:

Given,

$f\left(x\right)={\sin }^{-1}\sqrt{x-4}$.

$\begin{array}{rcl}f\left(x\right)& =& {\sin }^{-1}\sqrt{x-4}\\ & \Rightarrow & \sin \left[f\left(x\right)\right]=\sqrt{x-4}\\ & \Rightarrow & x-4={\sin }^2\left[f\left(x\right)\right]\end{array}$

Concept: The range of $x$ when ${\sin }^2\theta =x$ is $\left[0,1\right]$.

Therefore, range of $\left(x-4\right)$.

$$\begin{gathered} 0\le x-4\le 1 \\ \Rightarrow 4\le x\le 5 \end{gathered}$$

Hence, the correct option is A.