The domain of the function `f(x)=sin^(-1)log_(3)(x/3))` is

Options

A. [1, 9]
B. (-1, 9)
C. (-9, 1)
D. (-9, -1)

Answer

To find the domain of the function \( f(x) = \sin^{-1}(\log_{3}(x/3)) \), we need to ensure that the argument of the inverse sine function, \( \log_{3}(x/3) \), lies within the range of the sine inverse function. The range of \( \sin^{-1}(x) \) is \([-1, 1]\). Therefore, we need to solve the inequality:

\[ -1 \leq \log_{3}(x/3) \leq 1 \]

### Step 1: Solve the left side of the inequality
Starting with the left side:

\[ \log_{3}(x/3) \geq -1 \]

Using the property of logarithms, we can rewrite this as:

\[ x/3 \geq 3^{-1} \]

This simplifies to:

\[ x/3 \geq \frac{1}{3} \]

Multiplying both sides by 3 gives:

\[ x \geq 1 \]

### Step 2: Solve the right side of the inequality
Now, we solve the right side:

\[ \log_{3}(x/3) \leq 1 \]

Again, using the property of logarithms, we rewrite this as:

\[ x/3 \leq 3^{1} \]

This simplifies to:

\[ x/3 \leq 3 \]

Multiplying both sides by 3 gives:

\[ x \leq 9 \]

### Step 3: Combine the results
From the two parts, we have:

\[ 1 \leq x \leq 9 \]

Thus, the domain of the function \( f(x) = \sin^{-1}(\log_{3}(x/3)) \) is:

\[ [1, 9] \]

### Final Answer
The domain of the function is \( [1, 9] \).