Let's analyze the problem presented in the image.

Given Functions:

• $f(x) = \frac{-4x + 5}{x + 7}$
• $g(x) = \frac{4x - 9}{x - 6}$

Operations and Results:

1. Sum $(f + g)(x)$: $(f + g)(x) = \frac{-4x + 5}{x + 7} + \frac{4x - 9}{x - 6}$ To simplify the sum, find the least common denominator (LCD) of $(x+7)$ and $(x-6)$: $(f + g)(x) = \frac{(-4x + 5)(x - 6) + (4x - 9)(x + 7)}{(x + 7)(x - 6)}$ This expression simplifies after expanding the numerator.

2. Difference $(f - g)(x)$: $(f - g)(x) = \frac{-4x + 5}{x + 7} - \frac{4x - 9}{x - 6}$ Similarly, you can find the least common denominator and simplify: $(f - g)(x) = \frac{(-4x + 5)(x - 6) - (4x - 9)(x + 7)}{(x + 7)(x - 6)}$ In the image, this simplifies to: $(f - g)(x) = \frac{-8x^2 + 10x + 33}{(x + 7)(x - 6)}$

3. Product $(fg)(x)$: $(fg)(x) = \frac{-4x + 5}{x + 7} \times \frac{4x - 9}{x - 6}$ The numerator and denominator are multiplied: $(fg)(x) = \frac{(-4x + 5)(4x - 9)}{(x + 7)(x - 6)}$ This simplifies to: $(fg)(x) = \frac{-16x^2 + 56x - 45}{(x + 7)(x - 6)}$

Domains:

The domain for these functions is determined by the restrictions on the denominator, which cannot be zero.

• For $f(x) = \frac{-4x + 5}{x + 7}$, the domain restriction is $x \neq -7$.
• For $g(x) = \frac{4x - 9}{x - 6}$, the domain restriction is $x \neq 6$.

Thus, the domains of the combined functions are restricted by both denominators:

• Domain of $(f + g)(x)$, $(f - g)(x)$, and $(fg)(x)$: $x \neq -7$ and $x \neq 6$.

Would you like more details on any of these steps? Here are five related questions for further practice:

1. How do you expand and simplify rational expressions with different denominators?
2. What is the process for finding the least common denominator of two rational functions?
3. How would the result change if you added, subtracted, or multiplied two functions with the same denominator?
4. What are the steps for finding the domain of a rational function?
5. How can you check if your simplified result of a rational function operation is correct?

Tip: Always check for restrictions in the domain when working with rational functions—especially before simplifying!