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.