Question

The domain of ${\sin }^{-1}\left[{\log }_3\left(\frac{x}{3}\right)\right]$ is

• A. $\left[1,9\right]$
• B. $\left[-1,9\right]$
• C. $\left[-9,1\right]$
• D. $\left[-9,-1\right]$

Answer 1

The correct option is A

$\left[1,9\right]$

Explanation for the correct option:

Step 1: Finding the minimum and maximum value

Given $f\left(x\right)={\sin }^{-1}\left[{\log }_3\left(\frac{x}{3}\right)\right]$

We know that $-1\le \sin \theta \le 1$

so for inverse function, ${\sin }^{-1}\left(x\right)$ the value of $x$ will lies in the range of $-1\le x\le 1$

So for the given value of function we have,

$-1\le {\log }_3\left(\frac{x}{3}\right)\le 1$

We know that

$$\begin{gathered} {\log }_{a}x=b \\ x=a^b \end{gathered}$$

Hence,

$$\begin{gathered} {\log }_3\frac{x}{3}\ge -1 \\ \frac{x}{3}\ge 3^{-1} \\ x\ge 1 \end{gathered}$$

So, minimum value of $x$ is $1$

Similarly, we have

$$\begin{gathered} {\log }_3\frac{x}{3}\le 1 \\ \frac{x}{3}\le 3 \\ x\le 9 \end{gathered}$$

So,maximum value of $x$ is $9$

Step 2: Finding the domain.

So, the minimum and maximum value of $x$ forms the domain of the given function $f\left(x\right)={\sin }^{-1}\left[{\log }_3\left(\frac{x}{3}\right)\right]$

Hence, the domain of the function $f\left(x\right)$ is $\left[1,9\right]$

Therefore, option (A) is the correct answer.