Find $$ f+g, f-g, f g $$, and $$ \frac{f}{g} $$. Determine the domain for each function. $$ f(x)=x+1, g(x)=5 x^{2} $$ $$ (f+g)(x)=5 x^{2}+x+1 $$ (Simplify your answer.) What is the domain of $$ f+g $$ ? A. The domain of $$ f+g $$ is $$ (-\infty, \infty) $$. (Type your answer in interval notation.) C. The domain of $$ f+g $$ is $$ \{\quad $$. (Use a comma to separate answers as needed.) $$ (f-g)(x)=\square $$ (Simplify your answer.) C $$ f+g $$ is $$ \varnothing $$.

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Let's solve the problem step by step.

1. **Extract known conditions:**
   - $$ f(x) = x + 1 $$
   - $$ g(x) = 5x^2 $$

2. **Calculate $$ f + g $$:**
   $$
   (f + g)(x) = f(x) + g(x) = (x + 1) + (5x^2) = 5x^2 + x + 1
   $$
   This is already simplified.

3. **Determine the domain of $$ f + g $$:**
   - Both $$ f(x) $$ and $$ g(x) $$ are defined for all real numbers.
   - Therefore, the domain of $$ f + g $$ is $$ (-\infty, \infty) $$.

4. **Calculate $$ f - g $$:**
   $$
   (f - g)(x) = f(x) - g(x) = (x + 1) - (5x^2) = -5x^2 + x + 1
   $$
   This is also simplified.

5. **Determine the domain of $$ f - g $$:**
   - Similar to $$ f + g $$, both functions are defined for all real numbers.
   - Therefore, the domain of $$ f - g $$ is $$ (-\infty, \infty) $$.

6. **Calculate $$ f \cdot g $$:**
   $$
   (f \cdot g)(x) = f(x) \cdot g(x) = (x + 1)(5x^2) = 5x^3 + 5x^2
   $$
   This is simplified.

7. **Determine the domain of $$ f \cdot g $$:**
   - Again, both functions are defined for all real numbers.
   - Therefore, the domain of $$ f \cdot g $$ is $$ (-\infty, \infty) $$.

8. **Calculate $$ \frac{f}{g} $$:**
   $$
   \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x + 1}{5x^2}
   $$
   This is simplified.

9. **Determine the domain of $$ \frac{f}{g} $$:**
   - The function $$ g(x) = 5x^2 $$ cannot be zero, so we need to find where $$ g(x) = 0 $$:
     $$
     5x^2 = 0 \implies x = 0
     $$
   - Therefore, the domain of $$ \frac{f}{g} $$ is all real numbers except $$ x = 0 $$, which can be expressed in interval notation as $$ (-\infty, 0) \cup (0, \infty) $$.

### Summary of Results:
- $$ (f + g)(x) = 5x^2 + x + 1 $$
  - Domain: $$ (-\infty, \infty) $$
  
- $$ (f - g)(x) = -5x^2 + x + 1 $$
  - Domain: $$ (-\infty, \infty) $$

- $$ (f \cdot g)(x) = 5x^3 + 5x^2 $$
  - Domain: $$ (-\infty, \infty) $$

- $$ \left(\frac{f}{g}\right)(x) = \frac{x + 1}{5x^2} $$
  - Domain: $$ (-\infty, 0) \cup (0, \infty) $$
$$ (f + g)(x) = 5x^2 + x + 1 $$ with domain $$ (-\infty, \infty) $$. $$ (f - g)(x) = -5x^2 + x + 1 $$ with domain $$ (-\infty, \infty) $$. $$ (f \cdot g)(x) = 5x^3 + 5x^2 $$ with domain $$ (-\infty, \infty) $$. $$ \left(\frac{f}{g}\right)(x) = \frac{x + 1}{5x^2} $$ with domain $$ (-\infty, 0) \cup (0, \infty) $$.

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