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The problem asks to find the domain of the function \( f(x) = \sqrt{x - 8} \).

### **Step-by-Step Solution:**

1. **Understand the Function:**
   The function given is \( f(x) = \sqrt{x - 8} \). Since the square root function is only defined for non-negative values, the expression inside the square root, \( x - 8 \), must be greater than or equal to zero.

2. **Set Up the Inequality:**
   \[
   x - 8 \geq 0
   \]

3. **Solve the Inequality:**
   Add 8 to both sides to isolate \( x \):
   \[
   x \geq 8
   \]

4. **Interpret the Domain:**
   The domain of \( f(x) \) is all \( x \) values where \( x \geq 8 \).

5. **Express in Interval Notation:**
   The domain in interval notation is:
   \[
   [8, \infty)
   \]

### **Final Answer:**
The domain of \( f(x) = \sqrt{x - 8} \) is \( [8, \infty) \).

Would you like more details on this problem, or do you have any other questions?

### **Related Questions:**
1. What is the range of \( f(x) = \sqrt{x - 8} \)?
2. How would the domain change if the function were \( f(x) = \sqrt{8 - x} \)?
3. What is the domain of \( f(x) = \sqrt{(x - 8)(x + 2)} \)?
4. Can you find the domain and range of the function \( f(x) = \sqrt{4x - 16} \)?
5. How does adding a constant to the function \( f(x) = \sqrt{x - 8} \) affect its domain?

### **Tip:**
When working with square root functions, always check the expression inside the root to ensure it is non-negative, as this determines the domain of the function.

Find the domain of f(x) = √(x - 8). Use interval notation (use U for union when needed).

Understand how to find the domain of the square root function f(x) = √(x - 8). This problem involves setting up an inequality to determine where the function is defined, suitable for high school students in grades 9-12.

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