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It looks like you are working on a math problem related to functions and their domains, involving operations like addition, subtraction, multiplication, and division of functions.

From the image, we have the following information:

1. \( f(x) = x + 8 \)
2. \( g(x) = x - 8 \)

You are asked to find the following:

- \( (f + g)(x) \)
- \( (f - g)(x) \)
- \( (fg)(x) \)
- \( \frac{f}{g}(x) \)
- The domain of \( \frac{f}{g}(x) \)

Let’s go step by step.

### (a) \( (f + g)(x) \)
\[ (f + g)(x) = f(x) + g(x) = (x + 8) + (x - 8) = 2x \]

### (b) \( (f - g)(x) \)
\[ (f - g)(x) = f(x) - g(x) = (x + 8) - (x - 8) = x + 8 - x + 8 = 16 \]

### (c) \( (fg)(x) \)
\[ (fg)(x) = f(x) \cdot g(x) = (x + 8)(x - 8) = x^2 - 64 \]

### (d) \( \frac{f}{g}(x) \)
\[ \frac{f}{g}(x) = \frac{x + 8}{x - 8} \]

### Domain of \( \frac{f}{g}(x) \)

The domain of \( \frac{f}{g}(x) \) is all real numbers except where the denominator is zero. So we need to find where \( g(x) = x - 8 = 0 \).

Solving \( x - 8 = 0 \), we get \( x = 8 \).

Thus, the domain of \( \frac{f}{g}(x) \) is all real numbers except \( x = 8 \). In interval notation, the domain is:
\[ (-\infty, 8) \cup (8, \infty) \]

### Summary of Answers:

- (a) \( (f + g)(x) = 2x \)
- (b) \( (f - g)(x) = 16 \)
- (c) \( (fg)(x) = x^2 - 64 \)
- (d) \( \frac{f}{g}(x) = \frac{x + 8}{x - 8} \)
- Domain of \( \frac{f}{g}(x) \): \( (-\infty, 8) \cup (8, \infty) \)

Would you like more details or explanations on any step? Here are some follow-up questions for deeper exploration:

1. How can you verify the domain of a function involving division?
2. What happens to the domain if a function includes a square root or logarithm?
3. Can you describe how multiplication of functions affects their domain?
4. How would you solve a problem if \( f(x) \) and \( g(x) \) involved more complex expressions?
5. What are the key differences between operations on functions and operations on real numbers?

**Tip:** Always check for restrictions in the domain, particularly when dealing with division or square roots in functions.

Find the following: (f + g)(x), (f - g)(x), (fg)(x), f/g(x), and determine the domain of f/g(x). Given f(x) = x + 8 and g(x) = x - 8.

Learn how to perform operations on functions, including addition, subtraction, multiplication, and division of f(x) = x + 8 and g(x) = x - 8. This problem also covers how to find the domain of the quotient of two functions.

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