https://file.aiquickdraw.com/mathbot/file/1725209020121fvig5h6i.jpg

Let's analyze the asymptotes for both functions provided in the image:

### Function \( c \): \( f(x) = \frac{2x}{x^2 + 2} \)

#### 1. **Vertical Asymptotes:**
Vertical asymptotes occur where the denominator equals zero, provided the numerator is not zero at the same point.

For \( f(x) = \frac{2x}{x^2 + 2} \), set the denominator equal to zero:
\[
x^2 + 2 = 0
\]
\[
x^2 = -2
\]
Since \( x^2 = -2 \) has no real solution, there are **no vertical asymptotes**.

#### 2. **Horizontal Asymptotes:**
For horizontal asymptotes, we compare the degrees of the numerator and the denominator.

- Degree of the numerator (2x): 1
- Degree of the denominator (\(x^2 + 2\)): 2

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:
\[
y = 0
\]

### Summary for \( c \):
- **Vertical asymptotes:** None
- **Horizontal asymptote:** \( y = 0 \)

---

### Function \( d \): \( f(x) = \frac{2 - x - x^2}{x + 1} \)

#### 1. **Vertical Asymptotes:**
Set the denominator equal to zero:
\[
x + 1 = 0
\]
\[
x = -1
\]
So, there is a vertical asymptote at \( x = -1 \).

#### 2. **Horizontal or Slant Asymptotes:**
To determine the horizontal or slant asymptote, divide the numerator by the denominator.

First, rewrite the function in a more standard form:
\[
f(x) = \frac{-x^2 - x + 2}{x + 1}
\]

Let's perform polynomial long division of \( -x^2 - x + 2 \) by \( x + 1 \):

- Divide \( -x^2 \) by \( x \) to get \( -x \).
- Multiply \( -x \) by \( x + 1 \) to get \( -x^2 - x \).
- Subtract \( -x^2 - x \) from \( -x^2 - x + 2 \) to get \( 2 \).

This gives:
\[
f(x) = -x + 2 + \frac{0}{x+1}
\]

Thus, the slant asymptote is:
\[
y = -x + 2
\]

### Summary for \( d \):
- **Vertical asymptote:** \( x = -1 \)
- **Slant asymptote:** \( y = -x + 2 \)

Would you like any further details or have any questions?

Here are some related questions:
1. How can you identify whether a function has a slant or horizontal asymptote?
2. What steps are involved in performing polynomial division for more complex numerators?
3. Can a function have multiple vertical asymptotes?
4. How does the degree of the polynomial affect the type of asymptote?
5. What are the real-world implications of horizontal and slant asymptotes in modeling?

**Tip:** For rational functions, if the degree of the numerator is exactly one more than the degree of the denominator, always check for a slant asymptote by performing division.

Explore how to find asymptotes in rational functions with detailed examples and solutions. Learn about vertical and horizontal asymptotes, and how to determine them using polynomial division and degree comparison. Suitable for advanced high school students.

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