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Let's break down the problem step by step.

### The Given Functions:
1. \( s(x) = \frac{x - 5}{x^2 - 49} \)
2. \( v(x) = \sqrt{x + 7} \)

### The Task:
We are asked to compute the function \( \left( \frac{v}{s} \right)(x) \), which is the division of \( v(x) \) by \( s(x) \), and then determine its domain in interval notation.

#### Step 1: Compute \( \frac{v(x)}{s(x)} \)
\[
\frac{v(x)}{s(x)} = \frac{\sqrt{x + 7}}{\frac{x - 5}{x^2 - 49}} = \sqrt{x + 7} \times \frac{x^2 - 49}{x - 5}
\]
Since \( x^2 - 49 = (x - 7)(x + 7) \), this simplifies to:
\[
\frac{v(x)}{s(x)} = \frac{\sqrt{x + 7} \cdot (x - 7)(x + 7)}{x - 5}
\]

#### Step 2: Domain Restrictions
To find the domain, we need to check the following:
1. The square root function \( \sqrt{x + 7} \) requires \( x + 7 \geq 0 \), so \( x \geq -7 \).
2. The denominator \( s(x) \) should not be zero. We solve:
   \[
   x^2 - 49 = 0 \implies (x - 7)(x + 7) = 0 \implies x = 7 \text{ or } x = -7
   \]
   Therefore, \( x = 7 \) or \( x = -7 \) makes the denominator zero, and these points must be excluded from the domain.

### Domain in Interval Notation:
- The function is defined for \( x \geq -7 \), except at \( x = 7 \) and \( x = -7 \).
- So the domain is: 
  \[
  [-7, 7) \cup (7, \infty)
  \]

This is the simplified function and the domain for \( \frac{v(x)}{s(x)} \).

Would you like a further breakdown of any steps or details? Here are 5 related questions:

1. What happens if the domain were restricted further by the behavior of \( s(x) \)?
2. How would you find the vertical asymptotes of this function?
3. What is the behavior of the function as \( x \to \infty \)?
4. Can you explore the zeros of \( v(x) \) or \( s(x) \)?
5. How would graphing \( v(x) \) and \( s(x) \) help visualize the solution?

**Tip**: Always check for restrictions imposed by square roots and denominators when determining the domain.

Find the indicated function (v/s)(x) and write the domain in interval notation using radicals when necessary. Given functions are: s(x) = (x - 5) / (x^2 - 49) and v(x) = √(x + 7).

Learn how to compute the function (v/s)(x) by dividing the square root function v(x) = √(x + 7) by the rational function s(x) = (x - 5) / (x^2 - 49). The solution covers finding the domain in interval notation and understanding the restrictions imposed by both the square root and rational function.

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