The domain of
`f(x) = cos[log_(5)(sqrt(25-x^(2))/(3-x))]`

Options

A. `(-infty,3)`
B. `(3,5)`
C. [3,5)
D. `R-(3,5)`

Answer

To find the domain of the function \( f(x) = \cos\left(\log_{5}\left(\frac{\sqrt{25 - x^{2}}}{3 - x}\right)\right) \), we need to ensure that the expression inside the logarithm is defined and positive, as the logarithm is only defined for positive values.

### Step 1: Identify the conditions for the logarithm
The logarithm \( \log_{5}\left(\frac{\sqrt{25 - x^{2}}}{3 - x}\right) \) is defined if:
1. \( \frac{\sqrt{25 - x^{2}}}{3 - x} > 0 \)

### Step 2: Analyze the numerator and denominator
The numerator \( \sqrt{25 - x^{2}} \) is defined and non-negative when:
- \( 25 - x^{2} \geq 0 \)
- This implies \( -5 \leq x \leq 5 \).

The denominator \( 3 - x \) must be positive:
- \( 3 - x > 0 \)
- This implies \( x < 3 \).

### Step 3: Combine the conditions
Now we have two conditions:
1. \( -5 \leq x \leq 5 \)
2. \( x < 3 \)

To find the intersection of these two conditions, we need to consider both ranges:
- From \( -5 \leq x \leq 5 \), we can take \( x \) values from \( -5 \) to \( 5 \).
- From \( x < 3 \), we can take values from \( -5 \) to just below \( 3 \).

### Step 4: Determine the final domain
The intersection of these two conditions gives us:
- The final domain is \( -5 \leq x < 3 \).

### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[ x \in [-5, 3) \]