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Let's solve the given problems step by step.

The given functions are:
- \( f(x) = -3x^2 + x - 7 \)
- \( g(x) = 5x + 11 \)

### a) \( (f + g)(x) \)
This is the sum of the functions \( f(x) \) and \( g(x) \):
\[
(f + g)(x) = f(x) + g(x)
\]
\[
(f + g)(x) = (-3x^2 + x - 7) + (5x + 11)
\]
Now combine like terms:
\[
(f + g)(x) = -3x^2 + (x + 5x) + (-7 + 11)
\]
\[
(f + g)(x) = -3x^2 + 6x + 4
\]

### b) The domain of \( (f + g)(x) \)
Both \( f(x) \) and \( g(x) \) are polynomials, and polynomials have a domain of all real numbers. Therefore, the domain of \( (f + g)(x) \) is:
\[
\text{Domain} = (-\infty, \infty)
\]

### c) \( (f - g)(x) \)
This is the difference between the functions \( f(x) \) and \( g(x) \):
\[
(f - g)(x) = f(x) - g(x)
\]
\[
(f - g)(x) = (-3x^2 + x - 7) - (5x + 11)
\]
Distribute the negative sign to \( g(x) \):
\[
(f - g)(x) = -3x^2 + x - 7 - 5x - 11
\]
Now combine like terms:
\[
(f - g)(x) = -3x^2 + (x - 5x) + (-7 - 11)
\]
\[
(f - g)(x) = -3x^2 - 4x - 18
\]

### d) The domain of \( (f - g)(x) \)
As before, both \( f(x) \) and \( g(x) \) are polynomials, so the domain is all real numbers:
\[
\text{Domain} = (-\infty, \infty)
\]

### e) \( (f \cdot g)(x) \)
This is the product of the functions \( f(x) \) and \( g(x) \):
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
\[
(f \cdot g)(x) = (-3x^2 + x - 7)(5x + 11)
\]
Now expand the product using the distributive property:
\[
(f \cdot g)(x) = -3x^2(5x + 11) + x(5x + 11) - 7(5x + 11)
\]
Distribute each term:
\[
(f \cdot g)(x) = (-15x^3 - 33x^2) + (5x^2 + 11x) - (35x + 77)
\]
Now combine like terms:
\[
(f \cdot g)(x) = -15x^3 + (-33x^2 + 5x^2) + (11x - 35x) - 77
\]
\[
(f \cdot g)(x) = -15x^3 - 28x^2 - 24x - 77
\]

### f) The domain of \( (f \cdot g)(x) \)
Again, since both \( f(x) \) and \( g(x) \) are polynomials, the domain is all real numbers:
\[
\text{Domain} = (-\infty, \infty)
\]

Would you like a deeper explanation on any step, or do you have questions about this? Here are five related questions:

1. How do we find the domain of a rational function?
2. What is the sum of two functions in general terms?
3. How can the product of two functions be simplified?
4. What are the differences between polynomial and rational functions in terms of domain?
5. How can we find the domain of composite functions?

**Tip:** Always check for any restrictions on the domain of functions, such as division by zero or square roots of negative numbers.

Given f(x) = -3x^2 + x - 7 and g(x) = 5x + 11, find (f + g)(x), the domain of (f + g)(x), (f - g)(x), the domain of (f - g)(x), (f * g)(x), and the domain of (f * g)(x).

Learn how to find the sum, difference, and product of polynomial functions f(x) = -3x^2 + x - 7 and g(x) = 5x + 11. This problem covers algebraic concepts such as polynomial function operations and determining the domain of these operations. Suitable for students in Grades 9-11.

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