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Let's solve the given problem.

We are given two functions:
- \( f(x) = \frac{x}{x - 7} \)
- \( g(x) = \frac{8 - x}{8 + x} \)

We are tasked with finding and expressing \( f(x+1)g(x+1) \) as a rational function.

### Step 1: Find \( f(x+1) \)
Substitute \( x + 1 \) into the function \( f(x) \):

\[
f(x+1) = \frac{x+1}{(x+1) - 7} = \frac{x+1}{x - 6}
\]

### Step 2: Find \( g(x+1) \)
Substitute \( x + 1 \) into the function \( g(x) \):

\[
g(x+1) = \frac{8 - (x+1)}{8 + (x+1)} = \frac{8 - x - 1}{8 + x + 1} = \frac{7 - x}{9 + x}
\]

### Step 3: Multiply \( f(x+1) \) and \( g(x+1) \)
Now, multiply the two expressions:

\[
f(x+1)g(x+1) = \left( \frac{x+1}{x - 6} \right) \times \left( \frac{7 - x}{9 + x} \right)
\]

This simplifies to:

\[
f(x+1)g(x+1) = \frac{(x+1)(7 - x)}{(x - 6)(9 + x)}
\]

### Step 4: Simplify further (if possible)
We leave the product in this factored form, as it is already the simplest rational expression.

### Step 5: Identify restrictions
The denominator \( (x - 6)(9 + x) \) should not be zero, so we have the restrictions:

\[
x \neq 6, -9
\]

Thus, the final answer is:

\[
f(x+1)g(x+1) = \frac{(x+1)(7 - x)}{(x - 6)(9 + x)}, \text{ for } x \neq 6, -9
\]

Let me know if you want further details or clarifications! Here are some related questions to consider:

1. What happens to \( f(x)g(x) \) when \( x = 6 \) or \( x = -9 \)?
2. How can you factorize rational expressions like this one more easily?
3. What are the conditions for simplifying products of rational functions?
4. How do the zeros of a rational function affect its graph?
5. How can you find the vertical asymptotes for this function?

**Tip:** Always check for restrictions on \( x \) when dealing with rational functions to avoid division by zero.

Let f(x) = x / (x - 7) and g(x) = (8 - x) / (8 + x). Express f(x + 1)g(x + 1) as a rational function.

Learn how to simplify the rational function f(x + 1)g(x + 1), where f(x) = x / (x - 7) and g(x) = (8 - x) / (8 + x). This detailed step-by-step solution demonstrates how to express it as a rational function with domain restrictions.

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