https://file.aiquickdraw.com/mathbot/file/1725552053272zw0uubfb.jpg

Let's analyze the two functions in the image to find their slant asymptotes:

### Problem 10:
The given function is:

\[
f(x) = \frac{4}{x+1}
\]

To find the slant asymptote (if it exists), we check the degrees of the numerator and denominator. In this case, the degree of the numerator is 0 (since 4 is a constant), and the degree of the denominator is 1 (since \(x+1\) is linear). A slant asymptote occurs when the degree of the numerator is exactly 1 greater than the degree of the denominator, which is not the case here.

**Conclusion**: There is no slant asymptote for this function. Instead, there is a horizontal asymptote at \(y = 0\), because the degree of the denominator is larger than the degree of the numerator.

---

### Problem 11:
The given function is:

\[
g(x) = \frac{4x^2 + 4x - 3x + 12}{2x^2 - 6x + 1}
\]

The degrees of both the numerator and denominator are 2 (quadratic). A slant asymptote occurs when the degree of the numerator is exactly one greater than the degree of the denominator, but since they are of the same degree, we will not have a slant asymptote.

However, in such cases, we often have a **horizontal asymptote** determined by dividing the leading terms of the numerator and the denominator.

The leading term of the numerator is \(4x^2\) and the leading term of the denominator is \(2x^2\). Dividing these gives:

\[
\frac{4x^2}{2x^2} = 2
\]

**Conclusion**: There is a horizontal asymptote at \(y = 2\) for this function.

---

Would you like more details on these solutions, or do you have any further questions?

Here are five related questions to explore:
1. How do you identify slant asymptotes for rational functions?
2. What conditions must be met for a function to have a horizontal asymptote?
3. How does polynomial long division relate to finding slant asymptotes?
4. What happens if the degree of the numerator is smaller than the degree of the denominator?
5. How do you determine vertical asymptotes?

**Tip**: For rational functions, always compare the degrees of the numerator and denominator to quickly determine the type of asymptote.

Find the slant asymptote of the functions f(x) = 4/(x + 1) and g(x) = (4x^2 + 4x - 3x + 12)/(2x^2 - 6x + 1).

Learn how to determine slant and horizontal asymptotes for rational functions. Explore the functions f(x) = 4/(x + 1) and g(x) = (4x^2 + 4x - 3x + 12)/(2x^2 - 6x + 1) to find the asymptotes based on the degrees of their numerators and denominators. This is ideal for high school students in Grades 11-12.

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