c. $$ \frac{-11}{x}, \frac{7}{x-4}, \frac{x}{x^{2}-16} $$

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Certainly! Let's analyze each of the given rational expressions:

### c. $$ \frac{-11}{x}, \frac{7}{x-4}, \frac{x}{x^{2}-16} $$

#### 1. $$ \frac{-11}{x} $$

- **Simplification**: This expression is already in its simplest form.
  
- **Domain**:
  - The denominator $$ x $$ cannot be zero because division by zero is undefined.
  - **Domain**: All real numbers except $$ x = 0 $$.
  - **Notation**: $$ \mathbb{R} \setminus \{0\} $$ or $$ x \in (-\infty, 0) \cup (0, \infty) $$

- **Vertical Asymptote**:
  - At $$ x = 0 $$, there's a vertical asymptote.

- **Horizontal Asymptote**:
  - As $$ x \to \pm\infty $$, $$ \frac{-11}{x} \to 0 $$. Thus, the horizontal asymptote is $$ y = 0 $$.

#### 2. $$ \frac{7}{x-4} $$

- **Simplification**: This expression is already in its simplest form.
  
- **Domain**:
  - The denominator $$ x - 4 $$ cannot be zero.
  - **Domain**: All real numbers except $$ x = 4 $$.
  - **Notation**: $$ \mathbb{R} \setminus \{4\} $$ or $$ x \in (-\infty, 4) \cup (4, \infty) $$

- **Vertical Asymptote**:
  - At $$ x = 4 $$, there's a vertical asymptote.

- **Horizontal Asymptote**:
  - As $$ x \to \pm\infty $$, $$ \frac{7}{x-4} \to 0 $$. Thus, the horizontal asymptote is $$ y = 0 $$.

#### 3. $$ \frac{x}{x^{2}-16} $$

- **Simplification**:
  - Factor the denominator:
    $$
    x^{2} - 16 = (x - 4)(x + 4)
    $$
  - So, the expression becomes:
    $$
    \frac{x}{(x - 4)(x + 4)}
    $$
  - This is the simplified form; no further simplification is possible unless factoring the numerator reveals common factors, which it doesn't in this case.

- **Domain**:
  - The denominator $$ x^{2} - 16 $$ cannot be zero.
  - Solve $$ x^{2} - 16 = 0 $$:
    $$
    x^{2} = 16 \implies x = \pm 4
    $$
  - **Domain**: All real numbers except $$ x = 4 $$ and $$ x = -4 $$.
  - **Notation**: $$ \mathbb{R} \setminus \{-4, 4\} $$ or $$ x \in (-\infty, -4) \cup (-4, 4) \cup (4, \infty) $$

- **Vertical Asymptotes**:
  - At $$ x = 4 $$ and $$ x = -4 $$, there are vertical asymptotes.

- **Horizontal Asymptote**:
  - Compare the degrees of the numerator and the denominator.
    - Degree of numerator: 1
    - Degree of denominator: 2
  - Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $$ y = 0 $$.

### Summary

| Expression              | Domain                     | Vertical Asymptotes | Horizontal Asymptote |
|-------------------------|----------------------------|---------------------|----------------------|
| $$ \frac{-11}{x} $$     | $$ x \neq 0 $$             | $$ x = 0 $$         | $$ y = 0 $$          |
| $$ \frac{7}{x-4} $$     | $$ x \neq 4 $$             | $$ x = 4 $$         | $$ y = 0 $$          |
| $$ \frac{x}{x^{2}-16} $$| $$ x \neq \pm4 $$          | $$ x = \pm4 $$      | $$ y = 0 $$          |

### Graphical Behavior

- **Vertical Asymptotes**: These lines are where the function grows without bound (either positive or negative infinity). They occur at values of $$ x $$ that make the denominator zero.
  
- **Horizontal Asymptote**: As $$ x $$ approaches positive or negative infinity, the function approaches the horizontal asymptote. For all three functions, as $$ x $$ becomes very large or very small, the value of the function approaches zero.

- **Intercepts**:
  - **y-intercept**: Set $$ x = 0 $$ (if within the domain).
    - For $$ \frac{-11}{x} $$, $$ x = 0 $$ is not in the domain (no y-intercept).
    - For $$ \frac{7}{x-4} $$, when $$ x = 0 $$, $$ y = \frac{7}{-4} = -\frac{7}{4} $$.
    - For $$ \frac{x}{x^{2}-16} $$, when $$ x = 0 $$, $$ y = 0 $$.

- **x-intercept**: Set $$ y = 0 $$ and solve for $$ x $$.
    - For $$ \frac{-11}{x} $$, $$ \frac{-11}{x} = 0 $$ has no solution (no x-intercept).
    - For $$ \frac{7}{x-4} $$, $$ \frac{7}{x-4} = 0 $$ has no solution (no x-intercept).
    - For $$ \frac{x}{x^{2}-16} $$, $$ \frac{x}{x^{2}-16} = 0 $$ when $$ x = 0 $$.

### Additional Notes

- **Behavior Near Asymptotes**:
  - It's useful to analyze the sign of the function in the intervals determined by the vertical asymptotes to understand how the function behaves near these points.

- **Graph Sketching**:
  - When sketching the graph of these functions, plot the asymptotes, intercepts, and analyze the behavior in each interval defined by the asymptotes.

If you have a specific question or need further analysis on these expressions (such as solving equations involving them, integrating, differentiating, or graphing), feel free to ask!
Here are the simplified forms and key features of the given rational expressions: 1. **$$ \frac{-11}{x} $$** - **Domain**: All real numbers except $$ x = 0 $$. - **Vertical Asymptote**: $$ x = 0 $$. - **Horizontal Asymptote**: $$ y = 0 $$. 2. **$$ \frac{7}{x-4} $$** - **Domain**: All real numbers except $$ x = 4 $$. - **Vertical Asymptote**: $$ x = 4 $$. - **Horizontal Asymptote**: $$ y = 0 $$. 3. **$$ \frac{x}{x^{2}-16} $$** - **Domain**: All real numbers except $$ x = 4 $$ and $$ x = -4 $$. - **Vertical Asymptotes**: $$ x = 4 $$ and $$ x = -4 $$. - **Horizontal Asymptote**: $$ y = 0 $$. **Summary Table:** | Expression | Domain | Vertical Asymptotes | Horizontal Asymptote | |-------------------------|----------------------------|---------------------|----------------------| | $$ \frac{-11}{x} $$ | $$ x \neq 0 $$ | $$ x = 0 $$ | $$ y = 0 $$ | | $$ \frac{7}{x-4} $$ | $$ x \neq 4 $$ | $$ x = 4 $$ | $$ y = 0 $$ | | $$ \frac{x}{x^{2}-16} $$| $$ x \neq \pm4 $$ | $$ x = \pm4 $$ | $$ y = 0 $$ | These expressions have vertical asymptotes where the denominator is zero and a horizontal asymptote at $$ y = 0 $$ since the degree of the numerator is less than the degree of the denominator.

c. \( \frac{-11}{x}, \frac{7}{x-4}, \frac{x}{x^{2}-16} \)

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