Let's work through the operations on the given functions $f(x) = 3x - 2$ and $g(x) = 5x + 2$. Below are the basic operations we can perform with these functions:

1. Addition of Functions:

$(f + g)(x) = f(x) + g(x)$ $(f + g)(x) = (3x - 2) + (5x + 2) = 3x + 5x - 2 + 2 = 8x$ Thus, $(f + g)(x) = 8x$.

2. Subtraction of Functions:

$(f - g)(x) = f(x) - g(x)$ $(f - g)(x) = (3x - 2) - (5x + 2) = 3x - 5x - 2 - 2 = -2x - 4$ Thus, $(f - g)(x) = -2x - 4$.

3. Multiplication of Functions:

$(f \cdot g)(x) = f(x) \cdot g(x)$ $(f \cdot g)(x) = (3x - 2)(5x + 2)$ Using the distributive property: $(f \cdot g)(x) = 3x(5x + 2) - 2(5x + 2) = 15x^2 + 6x - 10x - 4 = 15x^2 - 4x - 4$ Thus, $(f \cdot g)(x) = 15x^2 - 4x - 4$.

4. Division of Functions (where $g(x) \neq 0$):

$\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}$ $\left( \frac{f}{g} \right)(x) = \frac{3x - 2}{5x + 2}$ Thus, $\left( \frac{f}{g} \right)(x) = \frac{3x - 2}{5x + 2}$, for $x \neq -\frac{2}{5}$ to avoid division by zero.

5. Composition of Functions:

Let's compute $f(g(x))$ and $g(f(x))$.

a) $f(g(x))$:

$f(g(x)) = f(5x + 2) = 3(5x + 2) - 2 = 15x + 6 - 2 = 15x + 4$ Thus, $f(g(x)) = 15x + 4$.

b) $g(f(x))$:

$g(f(x)) = g(3x - 2) = 5(3x - 2) + 2 = 15x - 10 + 2 = 15x - 8$ Thus, $g(f(x)) = 15x - 8$.

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Would you like more details or explanations on any of these operations?

Here are 5 follow-up questions to expand on this:

1. What happens when we find the inverse of both $f(x)$ and $g(x)$?
2. Can we find a real solution to $f(x) = g(x)$?
3. How does the composition $f(f(x))$ or $g(g(x))$ behave?
4. What is the domain and range of $f(x)$ and $g(x)$?
5. Can we solve $f(x)g(x) = 0$? If so, how?

Tip: When working with function composition, always apply the innermost function first.