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The problem asks for the domain and range of the function \( f - g \), where:

\[
f(x) = \sqrt{x + 5}, \quad g(x) = \sqrt{x - 5}
\]

### Step 1: Determine the domain of \( f(x) \) and \( g(x) \)

1. **Domain of \( f(x) = \sqrt{x + 5} \):**
   \[
   x + 5 \geq 0 \implies x \geq -5
   \]
   So, the domain of \( f(x) \) is \( x \geq -5 \).

2. **Domain of \( g(x) = \sqrt{x - 5} \):**
   \[
   x - 5 \geq 0 \implies x \geq 5
   \]
   So, the domain of \( g(x) \) is \( x \geq 5 \).

Since \( f - g \) exists only where both \( f(x) \) and \( g(x) \) are defined, the domain of \( f - g \) is the intersection of the two domains:
\[
\text{Domain of } f - g: x \geq 5
\]

### Step 2: Determine the range of \( f - g \)

- **For \( f(x) = \sqrt{x + 5} \)**:
  - Minimum value: \( f(5) = \sqrt{10} \)
  - As \( x \) increases, \( f(x) \) increases without bound, so the range of \( f(x) \) is \( [\sqrt{10}, \infty) \).

- **For \( g(x) = \sqrt{x - 5} \)**:
  - Minimum value: \( g(5) = 0 \)
  - As \( x \) increases, \( g(x) \) increases without bound, so the range of \( g(x) \) is \( [0, \infty) \).

- **Range of \( f - g \)**:
  - Minimum value: \( f(5) - g(5) = \sqrt{10} - 0 = \sqrt{10} \)
  - As \( x \) increases, \( f(x) - g(x) \) increases without bound.

Thus, the range of \( f - g \) is \( [\sqrt{10}, \infty) \).

### Final Answer
- **Domain**: \( x \geq 5 \)
- **Range**: \( y \geq \sqrt{10} \)

This corresponds to option **A** in the provided choices.

Would you like more details on any part of this explanation, or do you have any other questions?

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Here are some related questions to expand your understanding:

1. How do you find the domain and range of a function that involves square roots?
2. What is the difference between the union and intersection of domains?
3. How does the range of a function change when you subtract two functions?
4. How would the domain and range change if \( g(x) \) was a linear function instead of a square root function?
5. What are the implications if \( x \) in the square root functions is replaced with a quadratic expression?

**Tip:** When dealing with square roots, always remember that the expression inside the root must be non-negative to ensure real number outputs.

Find the domain and range for f - g, where f(x) = √(x + 5) and g(x) = √(x - 5).

Learn how to find the domain and range for the function f - g where f(x) = √(x + 5) and g(x) = √(x - 5). This problem explores the intersection of domains and calculating ranges for square root functions. Suitable for Grades 9-11.

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