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

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To solve the problem, we will find $$ f + g $$, $$ f - g $$, $$ f g $$, and $$ \frac{f}{g} $$ for the given functions $$ f(x) = 5x + 2 $$ and $$ g(x) = x - 6 $$. We will also determine the domain for each function.

### Step 1: Calculate $$ f + g $$

$$
f + g = (5x + 2) + (x - 6) = 5x + 2 + x - 6 = 6x - 4
$$

### Step 2: Calculate $$ f - g $$

$$
f - g = (5x + 2) - (x - 6) = 5x + 2 - x + 6 = 4x + 8
$$

### Step 3: Calculate $$ f g $$

$$
f g = (5x + 2)(x - 6) = 5x^2 - 30x + 2x - 12 = 5x^2 - 28x - 12
$$

### Step 4: Calculate $$ \frac{f}{g} $$

$$
\frac{f}{g} = \frac{5x + 2}{x - 6}
$$

### Step 5: Determine the domains

1. **Domain of $$ f + g $$**:
   - Since $$ f(x) $$ and $$ g(x) $$ are both polynomials, their sum $$ f + g $$ is also a polynomial. The domain is all real numbers.
   - Domain: $$ (-\infty, \infty) $$

2. **Domain of $$ f - g $$**:
   - Similar to $$ f + g $$, $$ f - g $$ is a polynomial. The domain is all real numbers.
   - Domain: $$ (-\infty, \infty) $$

3. **Domain of $$ f g $$**:
   - The product $$ f g $$ is also a polynomial, so the domain is all real numbers.
   - Domain: $$ (-\infty, \infty) $$

4. **Domain of $$ \frac{f}{g} $$**:
   - The function $$ \frac{f}{g} $$ is a rational function. The domain is all real numbers except where the denominator is zero.
   - Set $$ g(x) = 0 $$:
     $$
     x - 6 = 0 \implies x = 6
     $$
   - Therefore, the domain of $$ \frac{f}{g} $$ is all real numbers except $$ x = 6 $$.
   - Domain: $$ (-\infty, 6) \cup (6, \infty) $$

### Summary of Results

- $$ f + g = 6x - 4 $$
- $$ f - g = 4x + 8 $$
- $$ f g = 5x^2 - 28x - 12 $$
- $$ \frac{f}{g} = \frac{5x + 2}{x - 6} $$

### Domains

- Domain of $$ f + g $$: $$ (-\infty, \infty) $$
- Domain of $$ f - g $$: $$ (-\infty, \infty) $$
- Domain of $$ f g $$: $$ (-\infty, \infty) $$
- Domain of $$ \frac{f}{g} $$: $$ (-\infty, 6) \cup (6, \infty) $$
$$ f + g = 6x - 4 $$, $$ f - g = 4x + 8 $$, $$ f g = 5x^2 - 28x - 12 $$, and $$ \frac{f}{g} = \frac{5x + 2}{x - 6} $$. The domains are all real numbers for $$ f + g $$, $$ f - g $$, and $$ f g $$, and all real numbers except $$ x = 6 $$ for $$ \frac{f}{g} $$.

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