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The problem asks to find the following combinations of functions and determine their domains for \( f(x) = 4x + 1 \) and \( g(x) = x - 8 \):

1. \( f + g \)
2. \( f - g \)
3. \( f \cdot g \) (fg)
4. \( \frac{f}{g} \)

Let's go step by step:

### 1. \( f + g \)
\[
(f + g)(x) = f(x) + g(x) = (4x + 1) + (x - 8) = 5x - 7
\]
- **Domain**: Both \( f(x) \) and \( g(x) \) are defined for all real numbers \( x \), so the domain is \( (-\infty, \infty) \).

### 2. \( f - g \)
\[
(f - g)(x) = f(x) - g(x) = (4x + 1) - (x - 8) = 4x + 1 - x + 8 = 3x + 9
\]
- **Domain**: Again, both functions are defined for all real numbers, so the domain is \( (-\infty, \infty) \).

### 3. \( f \cdot g \) (fg)
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (4x + 1)(x - 8) = 4x^2 - 32x + x - 8 = 4x^2 - 31x - 8
\]
- **Domain**: Both functions are defined for all real numbers, so the domain is \( (-\infty, \infty) \).

### 4. \( \frac{f}{g} \)
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{4x + 1}{x - 8}
\]
- **Domain**: The denominator \( g(x) = x - 8 \) cannot be zero, so \( x \neq 8 \). The domain is \( (-\infty, 8) \cup (8, \infty) \).

### Summary of Domains:
- \( f + g \): \( (-\infty, \infty) \)
- \( f - g \): \( (-\infty, \infty) \)
- \( f \cdot g \): \( (-\infty, \infty) \)
- \( \frac{f}{g} \): \( (-\infty, 8) \cup (8, \infty) \)

Let me know if you'd like more details or have any questions!

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Here are 5 related questions to expand on this:
1. How do you determine the domain of a function in general?
2. What happens to the domain when you combine functions through division?
3. How would the domain change if \( f(x) \) had a restricted domain?
4. Can you find the inverse of \( f(x) \) and \( g(x) \)?
5. What happens to the domain of \( \frac{f}{g} \) if both \( f(x) \) and \( g(x) \) have restrictions?

**Tip:** Always check for restrictions in the denominator when dividing functions to avoid undefined values.

Find f + g, f - g, fg, and f/g. Determine the domain for each function for f(x) = 4x + 1 and g(x) = x - 8.

Learn how to find f + g, f - g, fg, and f/g for the functions f(x) = 4x + 1 and g(x) = x - 8. The problem covers algebraic concepts including the operations of functions and domain restrictions, suitable for Grades 10-12.

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