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.