https://file.aiquickdraw.com/mathbot/file/1729315884932oagzy12t.jpg

For part (b), we are asked to rewrite the rational function \( y = \frac{x}{x^2 + 1} \) into the form \( y = r(x) + \frac{s(x)}{q(x)} \), where \( r(x) \) is a polynomial part and \( \frac{s(x)}{q(x)} \) is the remainder. Additionally, we will identify asymptotes and roughly sketch the graph based on the given information.

### Step 1: Rewrite the function
In this case, the function \( y = \frac{x}{x^2 + 1} \) is already in its simplest form, and it cannot be divided further into a polynomial part \( r(x) \) because the degree of the denominator (2) is greater than the degree of the numerator (1). Therefore, \( r(x) = 0 \), and the whole function is already of the form \( \frac{s(x)}{q(x)} \) with:
- \( s(x) = x \)
- \( q(x) = x^2 + 1 \).

### Step 2: Asymptotes
For asymptotes, we look at the behavior of the function as \( x \to \infty \) or \( x \to -\infty \):
- **Horizontal asymptote:** Since the degree of the denominator is higher than the degree of the numerator, the function tends to 0 as \( x \to \infty \). Thus, there is a **horizontal asymptote at \( y = 0 \)**.
- **Vertical asymptote:** Vertical asymptotes occur where the denominator is zero, but \( x^2 + 1 = 0 \) has no real roots. Therefore, there are **no vertical asymptotes**.

### Step 3: Rough sketch of the graph
- The function has a horizontal asymptote at \( y = 0 \).
- The function is symmetric with respect to the y-axis because it is an odd function.
- At \( x = 0 \), the value of the function is \( y(0) = 0 \), meaning it passes through the origin.
- For large positive and negative values of \( x \), the function approaches 0.

This information suggests that the graph will approach the x-axis from both the left and right but will never cross the asymptote \( y = 0 \) at infinity.

Would you like further details or clarifications?

### 5 Related Questions:
1. How do you determine horizontal asymptotes for rational functions?
2. Why doesn't the function \( y = \frac{x}{x^2 + 1} \) have a vertical asymptote?
3. What happens to the graph of \( y = \frac{x}{x^2 + 1} \) as \( x \to \infty \)?
4. How do you find the intercepts of rational functions like \( y = \frac{x}{x^2 + 1} \)?
5. What are the symmetries of the function \( y = \frac{x}{x^2 + 1} \)?

### Tip:
To identify the horizontal asymptote of a rational function, always compare the degrees of the numerator and denominator.

Rewrite the rational function y = x / (x^2 + 1) into the form y = r(x) + s(x) / q(x), find its asymptotes, and sketch its graph.

Learn how to rewrite the function y = x / (x^2 + 1) in a specific form, identify its horizontal asymptote at y = 0, and sketch its graph. Understand key concepts about rational functions and asymptotic behavior, with a focus on function degrees.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation