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.