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!

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.