https://file.aiquickdraw.com/mathbot/file/17259343510254zaorgce.jpg

Let's analyze the given function:

\[
f(x) = \frac{x^2 + 2x + 1}{x + 1}
\]

### Step 1: Simplify the Function
First, factor the numerator:
\[
f(x) = \frac{(x + 1)(x + 1)}{x + 1} = \frac{(x + 1)^2}{x + 1}
\]

Now, simplify the function by canceling out the common factor \(x + 1\):
\[
f(x) = x + 1, \quad \text{for} \ x \neq -1
\]

However, since the original function had \(x + 1\) in the denominator, there is a **hole** at \(x = -1\).

### Step 2: Identify the Key Features of the Graph

1. **Domain**: The function is undefined at \(x = -1\), so the domain is:
   \[
   \text{Domain: } (-\infty, -1) \cup (-1, \infty)
   \]

2. **Vertical Asymptote**: There is no vertical asymptote because the denominator cancels out. Instead, there is a **hole** at \(x = -1\).

3. **Horizontal Asymptote**: Since the simplified form is a linear function \(f(x) = x + 1\), there is no horizontal asymptote.

4. **Intercepts**:
   - **x-intercept**: Set \(f(x) = 0\):
     \[
     x + 1 = 0 \quad \Rightarrow \quad x = -1
     \]
     However, since \(x = -1\) is where the hole is located, the function does not cross the x-axis.
   - **y-intercept**: Set \(x = 0\):
     \[
     f(0) = 0 + 1 = 1
     \]
     So, the y-intercept is \( (0, 1) \).

5. **Behavior Near the Hole**:
   As \(x\) approaches -1 from either side, the function approaches \(f(-1) = -1 + 1 = 0\), but it doesn't reach 0 due to the hole.

### Conclusion:
- **Domain**: \( (-\infty, -1) \cup (-1, \infty) \)
- **Hole**: at \(x = -1\)
- **No vertical or horizontal asymptotes**
- **Y-intercept**: \( (0, 1) \)

Would you like more details or have any other questions?

Here are some related questions to explore further:
1. How would the graph look with the hole at \(x = -1\)?
2. What are the implications of having a hole rather than a vertical asymptote?
3. How does the domain of the function affect its graph?
4. How can you identify holes in a function graphically?
5. What other types of discontinuities can occur in rational functions?

**Tip:** Always factor and simplify rational functions to identify key features like holes and asymptotes!

Determine each feature of the graph of the given function: f(x) = (x^2 + 2x + 1) / (x + 1)

Explore the graph features of the rational function f(x) = (x^2 + 2x + 1) / (x + 1). This solution covers key concepts like simplification, holes in the graph, intercepts, and asymptotes. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation