Graph all asymptotes of the rational functior $$ f(x)=\frac{x^{2}+7 x-3}{x+3} $$

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To analyze and graph all asymptotes of the rational function:

$$
f(x) = \frac{x^{2} + 7x - 3}{x + 3}
$$

we need to determine both **vertical** and **oblique (slant)** asymptotes.

### 1. Vertical Asymptote

**Vertical asymptotes** occur where the denominator of a rational function equals zero (as long as the numerator isn't zero at the same point, which would indicate a hole instead).

**Steps to find the vertical asymptote:**

Set the denominator equal to zero and solve for $$ x $$:

$$
x + 3 = 0 \quad \Rightarrow \quad x = -3
$$

**Conclusion:**

- **Vertical Asymptote:** $$ x = -3 $$

### 2. Oblique (Slant) Asymptote

Since the degree of the numerator (2) is **one degree higher** than the degree of the denominator (1), the function has an **oblique asymptote**. To find it, perform polynomial long division of the numerator by the denominator.

**Steps for Polynomial Long Division:**

Divide $$ x^{2} + 7x - 3 $$ by $$ x + 3 $$:

1. **Divide:** $$ x^{2} \div x = x $$
2. **Multiply:** $$ x \times (x + 3) = x^{2} + 3x $$
3. **Subtract:** 
   $$
   (x^{2} + 7x - 3) - (x^{2} + 3x) = 4x - 3
   $$
4. **Divide:** $$ 4x \div x = 4 $$
5. **Multiply:** $$ 4 \times (x + 3) = 4x + 12 $$
6. **Subtract:** 
   $$
   (4x - 3) - (4x + 12) = -15
   $$

The division yields:

$$
f(x) = x + 4 + \frac{-15}{x + 3}
$$

As $$ x $$ approaches infinity or negative infinity, the term $$ \frac{-15}{x + 3} $$ approaches 0. Therefore, the oblique asymptote is the linear part of the division result.

**Conclusion:**

- **Oblique Asymptote:** $$ y = x + 4 $$

### 3. Graphical Representation

While I can't provide a visual graph here, I can describe how the asymptotes and the graph of the function relate:

- **Vertical Asymptote ($$ x = -3 $$):** The graph approaches this vertical line but never touches or crosses it. As $$ x $$ approaches -3 from the left, $$ f(x) $$ tends to negative infinity, and as $$ x $$ approaches -3 from the right, $$ f(x) $$ tends to positive infinity.

- **Oblique Asymptote ($$ y = x + 4 $$):** As $$ x $$ becomes very large in the positive or negative direction, the graph of $$ f(x) $$ will get closer and closer to the line $$ y = x + 4 $$. This line acts as the "guide" for the end behavior of the function.

### 4. Summary of Asymptotes

- **Vertical Asymptote:** $$ x = -3 $$
- **Oblique Asymptote:** $$ y = x + 4 $$

### 5. Sketching the Graph (Description)

1. **Draw the vertical asymptote** as a dashed vertical line at $$ x = -3 $$.
2. **Draw the oblique asymptote** as a dashed line following $$ y = x + 4 $$.
3. **Plot key points:** Choose values of $$ x $$ around the vertical asymptote to determine the behavior of $$ f(x) $$ near $$ x = -3 $$.
4. **Plot the function:** Sketch the curve approaching both asymptotes. On one side of the vertical asymptote, the graph will head towards positive infinity, and on the other side, it will head towards negative infinity, all while following the oblique asymptote as $$ x $$ moves away from -3.

### 6. Example Behavior

- **For $$ x > -3 $$:** The function will approach the oblique asymptote $$ y = x + 4 $$ from below if the remainder is negative or from above if positive. Given the remainder term $$ \frac{-15}{x + 3} $$, as $$ x $$ becomes large positive, $$ f(x) $$ approaches $$ y = x + 4 $$ from below.

- **For $$ x < -3 $$:** Similarly, as $$ x $$ becomes large negative, $$ f(x) $$ approaches $$ y = x + 4 $$ from above.

By understanding these asymptotes, you can accurately sketch the graph of the rational function $$ f(x) $$.
The function $$ f(x) = \frac{x^{2} + 7x - 3}{x + 3} $$ has: - **Vertical Asymptote:** $$ x = -3 $$ - **Oblique Asymptote:** $$ y = x + 4 $$ These asymptotes help in sketching the graph of the function.

Graph all asymptotes of the rational functior \[ f(x)=\frac{x^{2}+7 x-3}{x+3} \]

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