The range of function `f (x) = log _(e) sqrt(4- x ^(2))` is given by

Options

A. `(0,oo)`
B. `(-oo, oo)`
C. `(-oo, log _(e) 2]`
D. ` (log _(e)2,oo)`

Answer

To find the range of the function \( f(x) = \log_e \sqrt{4 - x^2} \), we will follow these steps:

### Step 1: Determine the domain of the function
The expression inside the square root, \( 4 - x^2 \), must be non-negative for the square root to be defined. Therefore, we set up the inequality:
\[ 4 - x^2 \geq 0 \]
This simplifies to:
\[ x^2 \leq 4 \]
Taking the square root of both sides gives:
\[ -2 \leq x \leq 2 \]
Thus, the domain of \( f(x) \) is \( [-2, 2] \).

### Step 2: Analyze the expression under the square root
Next, we need to analyze the expression \( \sqrt{4 - x^2} \). The maximum value occurs when \( x^2 \) is minimized. The minimum value of \( x^2 \) in the domain is \( 0 \) (when \( x = 0 \)), giving:
\[ \sqrt{4 - 0} = \sqrt{4} = 2 \]
The minimum value occurs when \( x^2 \) is maximized, which happens at the endpoints of the domain \( x = -2 \) or \( x = 2 \):
\[ \sqrt{4 - 4} = \sqrt{0} = 0 \]

### Step 3: Determine the range of \( \sqrt{4 - x^2} \)
From the analysis, we find that \( \sqrt{4 - x^2} \) takes values from \( 0 \) to \( 2 \):
\[ 0 \leq \sqrt{4 - x^2} \leq 2 \]

### Step 4: Apply the logarithm function
Now we apply the logarithm function \( f(x) = \log_e \sqrt{4 - x^2} \). The logarithm function is defined for positive values, so we need to consider the range of \( \sqrt{4 - x^2} \):
- When \( \sqrt{4 - x^2} = 0 \), \( f(x) \) is undefined.
- When \( \sqrt{4 - x^2} = 2 \), we find:
\[ f(x) = \log_e 2 \]

### Step 5: Determine the range of \( f(x) \)
Since \( \sqrt{4 - x^2} \) approaches \( 0 \) but never reaches it, \( f(x) \) approaches \( -\infty \) as \( \sqrt{4 - x^2} \) approaches \( 0 \). Therefore, the range of \( f(x) \) is:
\[ (-\infty, \log_e 2] \]

### Final Answer
The range of the function \( f(x) = \log_e \sqrt{4 - x^2} \) is:
\[ (-\infty, \log_e 2] \]