The range of `y=log_(3)(9-x^(2))`

Options

A. `{1,2}`
B. `(-oo,2]`
C. `(-oo,2)`
D. none of these

Answer

To find the range of the function \( y = \log_{3}(9 - x^2) \), we will follow these steps:

### Step 1: Determine the domain of the function
The logarithmic function is defined only for positive arguments. Therefore, we need to ensure that \( 9 - x^2 > 0 \).

### Step 2: Solve the inequality
We can rewrite the inequality:
\[ 9 - x^2 > 0 \]
This can be rearranged to:
\[ x^2 < 9 \]
Taking the square root of both sides, we find:
\[ -3 < x < 3 \]
Thus, the domain of \( x \) is \( (-3, 3) \).

### Step 3: Analyze the behavior of the function at the endpoints of the domain
Next, we will evaluate the function at the endpoints of the domain:
- As \( x \) approaches \( -3 \):
\[ y = \log_{3}(9 - (-3)^2) = \log_{3}(9 - 9) = \log_{3}(0) \rightarrow -\infty \]
- As \( x \) approaches \( 3 \):
\[ y = \log_{3}(9 - (3)^2) = \log_{3}(9 - 9) = \log_{3}(0) \rightarrow -\infty \]

### Step 4: Find the maximum value of the function
To find the maximum value of \( y \), we can evaluate the function at \( x = 0 \):
\[ y = \log_{3}(9 - 0^2) = \log_{3}(9) = \log_{3}(3^2) = 2 \]

### Step 5: Determine the range of the function
From our analysis:
- As \( x \) approaches \( -3 \) or \( 3 \), \( y \) approaches \( -\infty \).
- The maximum value of \( y \) is \( 2 \) when \( x = 0 \).

Thus, the range of the function is:
\[ (-\infty, 2] \]

### Final Answer
The range of \( y = \log_{3}(9 - x^2) \) is \( (-\infty, 2] \).

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