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### Given functions:
\( f(x) = 3x - 7 \)  
\( g(x) = 6 - x \)

### (a) Find \( (f + g)(x) \):
We add \( f(x) \) and \( g(x) \).

\[
(f + g)(x) = f(x) + g(x) = (3x - 7) + (6 - x) = 3x - x - 7 + 6 = 2x - 1
\]

So, \( (f + g)(x) = 2x - 1 \).

### (b) Find \( (f - g)(x) \):
We subtract \( g(x) \) from \( f(x) \).

\[
(f - g)(x) = f(x) - g(x) = (3x - 7) - (6 - x) = 3x - 7 - 6 + x = 4x - 13
\]

So, \( (f - g)(x) = 4x - 13 \).

### (c) Find \( (fg)(x) \):
We multiply \( f(x) \) and \( g(x) \).

\[
(fg)(x) = f(x) \cdot g(x) = (3x - 7)(6 - x) = 3x(6 - x) - 7(6 - x) = 18x - 3x^2 - 42 + 7x = -3x^2 + 25x - 42
\]

So, \( (fg)(x) = -3x^2 + 25x - 42 \).

### (d) Find \( \left( \frac{f}{g} \right)(x) \):
We divide \( f(x) \) by \( g(x) \).

\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 7}{6 - x}
\]

### Domain of \( \frac{f}{g} \):
The domain of a rational function excludes values of \( x \) that make the denominator zero. So, we solve:

\[
6 - x = 0 \implies x = 6
\]

Thus, the domain is all real numbers except \( x = 6 \). In interval notation, the domain is:

\[
(-\infty, 6) \cup (6, \infty)
\]

### Summary of Answers:
(a) \( (f + g)(x) = 2x - 1 \)  
(b) \( (f - g)(x) = 4x - 13 \)  
(c) \( (fg)(x) = -3x^2 + 25x - 42 \)  
(d) \( \left( \frac{f}{g} \right)(x) = \frac{3x - 7}{6 - x} \)  
Domain: \( (-\infty, 6) \cup (6, \infty) \)

Would you like further details on any part of these solutions?

Here are 5 related questions for practice:
1. What is \( (g + f)(x) \) for different given functions?
2. How would you find \( (g - f)(x) \)?
3. What happens to the domain if both \( f(x) \) and \( g(x) \) involve quadratic terms?
4. How would you solve \( (g \cdot f)(x) \) for more complex functions?
5. What is the domain of \( \frac{f}{g} \) when both functions have multiple variables?

**Tip:** When finding domains of rational functions, always check when the denominator becomes zero.

Consider the following functions: f(x) = 3x - 7, g(x) = 6 - x. Find (f + g)(x), (f - g)(x), (fg)(x), and (f/g)(x). What is the domain of (f/g)?

Learn how to perform function operations including addition, subtraction, multiplication, and division, and how to determine the domain of a rational function. This problem involves algebraic manipulation and domain analysis, suitable for high school students.

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