Let's focus on question number 2, where the function given is:

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

To sketch the graph of this function, let's analyze its properties step-by-step:

1. Domain

The function $f(x) = \frac{1}{x^2 + 1}$ is defined for all real values of $x$ because $x^2 + 1$ is always positive (it's never zero). Therefore, the domain is all real numbers, $x \in \mathbb{R}$.

2. Range

Since $x^2 + 1 \geq 1$ for all real $x$, $f(x)$ will always be between 0 and 1: $0 < f(x) \leq 1$ Thus, the range of the function is $0 < f(x) \leq 1$.

3. Asymptotes

There is no vertical asymptote because the function is defined for all $x$. However, as $x$ approaches infinity or negative infinity, $f(x)$ approaches 0. This means there is a horizontal asymptote at $y = 0$.

4. Symmetry

Since $f(-x) = f(x)$, the function is even, meaning the graph is symmetric with respect to the y-axis.

5. Behavior and Key Points

• At $x = 0$: $f(0) = \frac{1}{0^2 + 1} = 1$.
• As $x \to \infty$ or $x \to -\infty$, $f(x) \to 0$.

Sketch Summary

1. The graph has a maximum point at $(0, 1)$.
2. It approaches the x-axis (y = 0) as $x$ becomes very large in either direction.
3. The graph is symmetric about the y-axis.

Would you like further details on any of these steps or a graphical sketch of the function?

Additional Questions

1. What is the derivative of $f(x) = \frac{1}{x^2 + 1}$, and how does it affect the shape of the graph?
2. How does this function compare to $f(x) = \frac{1}{x^2}$ in terms of asymptotes and behavior?
3. How would the range of $f(x) = \frac{1}{x^2 + 1}$ change if the denominator were $x^2 + 2$?
4. What would be the effect of adding a constant to the function, like $f(x) = \frac{1}{x^2 + 1} + 1$?
5. How does the symmetry of $f(x) = \frac{1}{x^2 + 1}$ help in sketching the graph quickly?

Tip

For rational functions like $f(x) = \frac{1}{x^2 + 1}$, always check the domain and asymptotes first. This will provide a clear framework for sketching the graph.