Let's start by analyzing the given functions and solving the operations:

We have:

• $f(x) = \sqrt{3x - 2}$
• $g(x) = 3x + 6$

We are asked to find the following:

1. $(f + g)(x)$

This is simply the sum of $f(x)$ and $g(x)$: $(f + g)(x) = f(x) + g(x) = \sqrt{3x - 2} + (3x + 6)$

Now, for the domain:

• The function $f(x) = \sqrt{3x - 2}$ requires $3x - 2 \geq 0$, which gives: $x \geq \frac{2}{3}$
• The function $g(x) = 3x + 6$ has no restrictions on $x$ (its domain is $(-\infty, \infty)$).

Therefore, the domain of $(f + g)(x)$ is: $D = \left[ \frac{2}{3}, \infty \right)$

2. $(f - g)(x)$

This is the difference between $f(x)$ and $g(x)$: $(f - g)(x) = f(x) - g(x) = \sqrt{3x - 2} - (3x + 6)$

For the domain, it is the same as for $f(x)$: $D = \left[ \frac{2}{3}, \infty \right)$

3. $(f \cdot g)(x)$

This is the product of $f(x)$ and $g(x)$: $(fg)(x) = f(x) \cdot g(x) = \sqrt{3x - 2} \cdot (3x + 6)$

For the domain, we again use the domain of $f(x)$, since $g(x)$ has no restrictions: $D = \left[ \frac{2}{3}, \infty \right)$

4. $\left( \frac{f}{g} \right)(x)$

This is the quotient of $f(x)$ and $g(x)$: $\left( \frac{f}{g} \right)(x) = \frac{\sqrt{3x - 2}}{3x + 6}$

For the domain, two conditions must be met:

1. $3x - 2 \geq 0$ (so $f(x)$ is defined), which gives $x \geq \frac{2}{3}$.
2. $3x + 6 \neq 0$ (so the denominator is non-zero), which gives $x \neq -2$.

Thus, the domain is: $D = \left[ \frac{2}{3}, \infty \right) \setminus \{-2\}$

Summary:

1. $(f + g)(x) = \sqrt{3x - 2} + 3x + 6$
   • Domain: $\left[ \frac{2}{3}, \infty \right)$
2. $(f - g)(x) = \sqrt{3x - 2} - (3x + 6)$
   • Domain: $\left[ \frac{2}{3}, \infty \right)$
3. $(fg)(x) = \sqrt{3x - 2} \cdot (3x + 6)$
   • Domain: $\left[ \frac{2}{3}, \infty \right)$
4. $\left( \frac{f}{g} \right)(x) = \frac{\sqrt{3x - 2}}{3x + 6}$
   • Domain: $\left[ \frac{2}{3}, \infty \right) \setminus \{-2\}$

Let me know if you'd like further clarifications or have any other questions!

Related Questions:

1. How do we simplify radical functions like $f(x) = \sqrt{3x - 2}$?
2. What are the general rules for finding the domain of radical functions?
3. Why do we exclude $x = -2$ from the domain of $\frac{f}{g}(x)$?
4. How do sum and difference of functions affect their domains?
5. What happens if both functions have domain restrictions in a function operation?

Tip:

Always check for restrictions in both the numerator and denominator when dealing with quotient functions to avoid division by zero!