To solve and analyze the function $f(x) = \frac{x}{x^2 + 1}$, let's break it down:

1. Domain:

The function is defined for all real values of $x$ because the denominator $x^2 + 1$ is always positive for any real $x$, so there are no restrictions (i.e., no division by zero issues).

Domain: $(-\infty, \infty)$

2. Symmetry:

Check if the function is even or odd:

• A function is even if $f(-x) = f(x)$
• A function is odd if $f(-x) = -f(x)$

Let's calculate $f(-x)$: $f(-x) = \frac{-x}{(-x)^2 + 1} = \frac{-x}{x^2 + 1}$ Since $f(-x) = -f(x)$, the function is odd.

3. Asymptotes:

• Vertical asymptotes: There are no vertical asymptotes because the denominator $x^2 + 1$ never equals zero.
• Horizontal asymptotes: As $x \to \infty$ or $x \to -\infty$, the function behaves as: $\lim_{x \to \infty} \frac{x}{x^2 + 1} = 0$ Thus, the horizontal asymptote is $y = 0$.

4. Critical points and local extrema:

To find critical points, we differentiate the function: $f'(x) = \frac{(x^2 + 1) \cdot 1 - x \cdot 2x}{(x^2 + 1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$ Setting $f'(x) = 0$ to find critical points: $1 - x^2 = 0 \quad \Rightarrow \quad x^2 = 1 \quad \Rightarrow \quad x = \pm 1$

So, the critical points are $x = 1$ and $x = -1$.

• At $x = 1$, check the second derivative to determine if it's a local minimum or maximum.
• At $x = -1$, do the same.

5. Behavior at critical points:

You can further analyze the behavior at $x = 1$ and $x = -1$ by plugging these values into the second derivative or examining the sign changes of $f'(x)$.

Would you like me to dive deeper into these critical points or any other part of the analysis?

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Here are five related questions:

1. How do you find the second derivative of $f(x)$ for concavity analysis?
2. What is the significance of an odd function in terms of graph symmetry?
3. How do we determine whether the critical points are maxima, minima, or saddle points?
4. How does the horizontal asymptote affect the long-term behavior of the function?
5. How would we apply this analysis to a different rational function?

Tip: Always check the domain and asymptotes first when analyzing rational functions for important insights about their behavior.