Find $$ f+g, f-g, f g $$ and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=5 x+2, g(x)=x-6 $$ c. The domain of $$ f+g $$ is $$ \varnothing $$. $$ (f-g)(x)=4 x+8 $$ (Simplify your answer.) What is the domain of $$ f-g $$ ? A. The domain of $$ f-g $$ is \{ . (Use a comma to separate answers as needed.) B. The domain of $$ f-g $$ is . The dompe your answer in interval hof $$ f-g $$ is $$ \varnothing $$.

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Let's work through each part of the problem step by step.

### Given Functions:
- $$ f(x) = 5x + 2 $$
- $$ g(x) = x - 6 $$

### Operations on $$ f $$ and $$ g $$:

1. **Sum ($$ f + g $$)**:
   $$
   (f + g)(x) = f(x) + g(x) = (5x + 2) + (x - 6) = 6x - 4
   $$
   **Domain**: Both $$ f $$ and $$ g $$ are linear functions defined for all real numbers.
   $$
   \text{Domain of } (f + g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty)
   $$

2. **Difference ($$ f - g $$)**:
   $$
   (f - g)(x) = f(x) - g(x) = (5x + 2) - (x - 6) = 4x + 8
   $$
   **Domain**: Both $$ f $$ and $$ g $$ are defined for all real numbers.
   $$
   \text{Domain of } (f - g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty)
   $$

3. **Product ($$ f \cdot g $$)**:
   $$
   (f \cdot g)(x) = f(x) \times g(x) = (5x + 2)(x - 6) = 5x^2 - 30x + 2x - 12 = 5x^2 - 28x - 12
   $$
   **Domain**: Both $$ f $$ and $$ g $$ are defined for all real numbers.
   $$
   \text{Domain of } (f \cdot g) = \mathbb{R} \quad \text{or} \quad (-\infty, \infty)
   $$

4. **Quotient ($$ \frac{f}{g} $$)**:
   $$
   \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{5x + 2}{x - 6}
   $$
   **Domain**: The denominator $$ g(x) = x - 6 $$ cannot be zero.
   $$
   x - 6 \neq 0 \quad \Rightarrow \quad x \neq 6
   $$
   $$
   \text{Domain of } \left(\frac{f}{g}\right) = \mathbb{R} \setminus \{6\} \quad \text{or} \quad (-\infty, 6) \cup (6, \infty)
   $$

### Addressing Your Specific Questions:

#### Part c: Determining the Domain of $$ f + g $$
You mentioned:
> c. The domain of $$ f + g $$ is $$ \varnothing $$.

**Correction**:
The domain of $$ f + g $$ is **all real numbers**, not the empty set $$ \varnothing $$.

#### Determining the Domain of $$ f - g $$:

Given:
$$
(f - g)(x) = 4x + 8
$$
This is a linear function, which is defined for all real numbers.

**A. Domain as a Set**:
$$
\text{Domain of } (f - g) = \mathbb{R}
$$
If you need to list specific elements (though unnecessary here), it would include all real numbers. Since it's all-encompassing, no specific values need to be listed.

**B. Domain in Interval Notation**:
$$
\text{Domain of } (f - g) = (-\infty, \infty)
$$

#### Summary of Domains:

| Function          | Domain                        |
|-------------------|-------------------------------|
| $$ f + g $$       | $$ \mathbb{R} $$ or $$ (-\infty, \infty) $$ |
| $$ f - g $$       | $$ \mathbb{R} $$ or $$ (-\infty, \infty) $$ |
| $$ f \cdot g $$   | $$ \mathbb{R} $$ or $$ (-\infty, \infty) $$ |
| $$ \frac{f}{g} $$ | $$ \mathbb{R} \setminus \{6\} $$ or $$ (-\infty, 6) \cup (6, \infty) $$ |

### Final Answers for the Domain of $$ f - g $$:
- **A.** The domain of $$ f - g $$ is $$ \mathbb{R} $$ (all real numbers).
- **B.** The domain of $$ f - g $$ in interval notation is $$ (-\infty, \infty) $$.
The domain of $$ f - g $$ is all real numbers, $$ \mathbb{R} $$, or in interval notation, $$ (-\infty, \infty) $$.

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