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 the expression inside the logarithm is greater than zero:

\[ 9 - x^2 > 0 \]

### Step 2: Solve the inequality
To solve the inequality \( 9 - x^2 > 0 \):

\[ 9 > x^2 \]

Taking the square root of both sides, we find:

\[ -\sqrt{9} < x < \sqrt{9} \]

This simplifies to:

\[ -3 < x < 3 \]

### Step 3: Identify the maximum value of \( 9 - x^2 \)
The function \( 9 - x^2 \) is a downward-opening parabola. The maximum value occurs at the vertex, which is at \( x = 0 \):

\[ 9 - 0^2 = 9 \]

### Step 4: Calculate the maximum value of \( y \)
Now, substituting \( x = 0 \) into the function:

\[ y = \log_3(9) \]

Since \( 9 = 3^2 \), we can express this as:

\[ y = \log_3(3^2) = 2 \]

### Step 5: Determine the minimum value of \( y \)
As \( x \) approaches the boundaries of the domain (\(-3\) and \(3\)), \( 9 - x^2 \) approaches \( 0 \). Therefore, \( y \) approaches:

\[ y = \log_3(0) \to -\infty \]

### Step 6: Conclude the range of \( y \)
Thus, the range of \( y \) is:

\[ (-\infty, 2] \]

### Final Answer:
The range of \( y = \log_3(9 - x^2) \) is \( (-\infty, 2] \).

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