To find the domain of the function \( f(v) = \sqrt{v^2 - 2v - 8} \), we need to ensure that the expression inside the square root is non-negative. This is because the square root of a negative number is not defined in the set of real numbers.

### Step 1: Set the expression inside the square root greater than or equal to zero.
\[
v^2 - 2v - 8 \geq 0
\]

### Step 2: Solve the quadratic inequality.
First, solve the corresponding quadratic equation:
\[
v^2 - 2v - 8 = 0
\]
Factor the quadratic expression:
\[
(v - 4)(v + 2) = 0
\]
This gives the solutions:
\[
v = 4 \quad \text{or} \quad v = -2
\]

### Step 3: Determine the intervals.
The solutions \( v = 4 \) and \( v = -2 \) divide the real line into three intervals:
1. \( (-\infty, -2) \)
2. \( (-2, 4) \)
3. \( (4, \infty) \)

### Step 4: Test each interval.
- For \( v \in (-\infty, -2) \), pick \( v = -3 \):
  \[
  (-3)^2 - 2(-3) - 8 = 9 + 6 - 8 = 7 \quad (\text{positive})
  \]
- For \( v \in (-2, 4) \), pick \( v = 0 \):
  \[
  0^2 - 2(0) - 8 = -8 \quad (\text{negative, not allowed})
  \]
- For \( v \in (4, \infty) \), pick \( v = 5 \):
  \[
  5^2 - 2(5) - 8 = 25 - 10 - 8 = 7 \quad (\text{positive})
  \]

### Step 5: Include boundary points.
Since \( v = -2 \) and \( v = 4 \) make the expression zero, they are included in the domain.

### Final Answer:
The domain of \( f(v) = \sqrt{v^2 - 2v - 8} \) is:
\[
v \in (-\infty, -2] \cup [4, \infty)
\]

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### Related Questions:
1. How do you solve inequalities involving square roots?
2. What is the difference between solving quadratic equations and inequalities?
3. How do you find the domain of a rational function?
4. How does factoring help solve quadratic inequalities?
5. What other methods can you use to solve quadratic inequalities?

**Tip:** Always test intervals between the critical points of a quadratic inequality to determine where the expression is positive or negative.

This detailed solution explains how to find the domain of the function sqrt(v^2 - 2v - 8) by solving quadratic inequalities. The problem covers important algebraic concepts like quadratic equations and function domains. Suitable for high school students in Grades 9-11.

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