Find the domain of the following functions:
`(1)/(sqrt(25-x^(2)))`

Options

A. `x gt 5 or x lt -5`
B. `x gt 4 or x lt -5`
C. `x gt -5 or x lt 5`
D. `x gt 3 or x lt -5`

Answer

To find the domain of the function \( f(x) = \frac{1}{\sqrt{25 - x^2}} \), we need to ensure that the expression inside the square root is positive and that the denominator is not equal to zero. Here are the steps to find the domain:

### Step 1: Set the expression inside the square root greater than zero
We start with the inequality:
\[ 25 - x^2 > 0 \]

### Step 2: Rearrange the inequality
Rearranging gives us:
\[ 25 > x^2 \]
or equivalently,
\[ x^2 < 25 \]

### Step 3: Take the square root of both sides
Taking the square root of both sides, we find:
\[ -\sqrt{25} < x < \sqrt{25} \]
which simplifies to:
\[ -5 < x < 5 \]

### Step 4: Express the domain in interval notation
The solution to the inequality tells us that \( x \) must be in the interval:
\[ (-5, 5) \]

### Step 5: Check for values that make the denominator zero
We also need to ensure that the denominator is not zero. The denominator is zero when:
\[ \sqrt{25 - x^2} = 0 \Rightarrow 25 - x^2 = 0 \Rightarrow x^2 = 25 \Rightarrow x = 5 \text{ or } x = -5 \]
Since these points are not included in the interval, we confirm that they are excluded from the domain.

### Final Domain
Thus, the domain of the function \( f(x) = \frac{1}{\sqrt{25 - x^2}} \) is:
\[ \text{Domain: } (-5, 5) \]