The problem asks to determine the domain of the functions $f(x)$ and $g(x)$, and perform operations on them such as $f+g$, $f-g$, $f \cdot g$, and $\frac{f}{g}$, followed by determining the domain for each of these operations.

Given:

• $f(x) = 4x - 9$
• $g(x) = x - 3$

We are tasked with determining the domain of $f + g$.

Step 1: Add the functions $f(x)$ and $g(x)$

$(f + g)(x) = f(x) + g(x) = (4x - 9) + (x - 3) = 5x - 12$

Step 2: Find the domain of $f + g$

Since both $f(x)$ and $g(x)$ are linear functions (polynomials), their domains are all real numbers. Adding two polynomials doesn't introduce any restrictions (like division by zero or square roots of negative numbers), so the domain of $f + g$ is also all real numbers, denoted as $(-\infty, \infty)$.

Answer:

The domain of $f + g$ is: $\boxed{(-\infty, \infty)}$

The correct choice here would be option B.

Would you like to go over the domains for other operations like $f - g$, $f \cdot g$, and $\frac{f}{g}$?

Here are 5 relative questions for practice:

1. What is the domain of $f - g$ and why?
2. How do you determine the domain of $f \cdot g$?
3. What are the restrictions on the domain for $\frac{f}{g}$?
4. How does the degree of polynomials affect their domains?
5. How do we handle domain restrictions when combining functions?

Tip: When finding the domain of a quotient $\frac{f}{g}$, check that the denominator $g(x)$ does not equal zero, as division by zero is undefined.