https://file.aiquickdraw.com/mathbot/file/1729139481186wam5jdbz.jpg

The image contains two rational functions, and the task is to:

- Sketch the graph.
- Determine the **domain**, **range**, and **asymptotes** (vertical, horizontal, and oblique) for each function.

### Function 1: \( f(x) = \frac{x - 6}{x + 2} \)

#### Domain:
The domain consists of all values of \(x\) except where the denominator is zero. The denominator is zero when:
\[
x + 2 = 0 \implies x = -2
\]
Thus, the domain is \( x \in (-\infty, -2) \cup (-2, \infty) \).

#### Vertical Asymptote:
A vertical asymptote occurs where the denominator is zero (if it doesn’t cancel with the numerator). Here, at \( x = -2 \), there is a vertical asymptote.

#### Horizontal Asymptote:
The horizontal asymptote is determined by the degrees of the numerator and denominator:
- Since both the numerator and denominator are of degree 1, the horizontal asymptote is the ratio of the leading coefficients.
\[
y = \frac{1}{1} = 1
\]

#### Oblique Asymptote:
There is **no oblique asymptote** because the degrees of the numerator and denominator are the same.

#### Range:
As \(x\) approaches infinity or negative infinity, \(f(x)\) approaches the horizontal asymptote \(y = 1\). However, the function can take all values except \(y = 1\), which it approaches asymptotically.

Thus, the range is \( y \in (-\infty, 1) \cup (1, \infty) \).

---

### Function 2: \( f(x) = \frac{x^2 + 5x + 4}{x - 3} \)

#### Domain:
The domain consists of all values of \(x\) except where the denominator is zero:
\[
x - 3 = 0 \implies x = 3
\]
Thus, the domain is \( x \in (-\infty, 3) \cup (3, \infty) \).

#### Vertical Asymptote:
A vertical asymptote occurs where the denominator is zero, so there is a vertical asymptote at \( x = 3 \).

#### Horizontal Asymptote:
Since the degree of the numerator (2) is greater than the degree of the denominator (1), there is **no horizontal asymptote**. Instead, there will be an **oblique asymptote**.

#### Oblique Asymptote:
To find the oblique asymptote, perform polynomial long division:
\[
\frac{x^2 + 5x + 4}{x - 3}
\]
Dividing \(x^2 + 5x + 4\) by \(x - 3\) gives a quotient of \(x + 8\), with a remainder. Therefore, the oblique asymptote is the linear function \(y = x + 8\).

#### Range:
Since the graph has an oblique asymptote and there is no horizontal asymptote, the range is all real numbers, \( y \in (-\infty, \infty) \).

---

Would you like more details or a sketch of the graphs? Here are some additional questions to deepen your understanding:

1. How do you graph rational functions and locate vertical asymptotes?
2. What is the impact of the numerator's degree on the asymptotes?
3. How do you perform polynomial long division to find oblique asymptotes?
4. What happens to the graph near the vertical asymptote?
5. How does the horizontal asymptote affect the graph for large values of \(x\)?

**Tip:** Always check for removable discontinuities by factoring the numerator and denominator!

Sketch the graph and determine the domain, range, vertical, horizontal, and oblique asymptotes for the functions: 1. f(x) = (x - 6)/(x + 2) and 2. f(x) = (x^2 + 5x + 4)/(x - 3).

Learn how to sketch the graphs of the rational functions f(x) = (x - 6)/(x + 2) and f(x) = (x^2 + 5x + 4)/(x - 3). This detailed explanation covers the domain, range, and identification of vertical, horizontal, and oblique asymptotes. Ideal for high school students in Grades 11-12.

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