The function given in the image is:

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

Let us analyze and solve some relevant properties of this function. Below are steps you might be interested in:

1. Domain of $f(x)$:

The function is undefined when the denominator equals zero: $x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1.$ Thus, the domain of $f(x)$ is: $x \in \mathbb{R} \setminus \{-1, 1\}.$

2. Asymptotes:

• Vertical Asymptotes: Occur at points where the denominator is zero, i.e., $x = \pm 1$.
• Horizontal Asymptote: For large values of $x$ ($|x| \to \infty$): $\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{x^2}{x^2 - 1} = \frac{x^2}{x^2} = 1.$ Similarly, as $x \to -\infty$, the result is the same: $\lim_{x \to -\infty} f(x) = 1.$ Thus, the horizontal asymptote is $y = 1$.

3. Zeros of $f(x)$:

To find zeros, set the numerator to zero: $x^2 = 0 \implies x = 0.$ Thus, $f(x)$ has a zero at $x = 0$.

4. Behavior Near Vertical Asymptotes:

To examine the behavior near $x = 1$ and $x = -1$:

• As $x \to 1^+$, $f(x) \to +\infty$ (denominator approaches 0 from positive side).
• As $x \to 1^-$, $f(x) \to -\infty$.
• Similarly, as $x \to -1^+$, $f(x) \to -\infty$, and as $x \to -1^-$, $f(x) \to +\infty$.

5. Graphical Sketch:

The function has:

• A zero at $x = 0$.
• Vertical asymptotes at $x = \pm 1$.
• Horizontal asymptote at $y = 1$.

Would you like a graphical representation or deeper exploration of properties like intervals of increase/decrease, concavity, or further calculus-based analysis?

Here are 5 related questions to deepen understanding:

1. What is the derivative of $f(x)$, and what does it reveal about increasing/decreasing intervals?
2. How does $f(x)$ behave near $x = 0$? What is its slope at this point?
3. Can $f(x)$ have oblique asymptotes, and why or why not?
4. How would modifying the numerator to $x^3$ change the function's graph?
5. What are the concavity and inflection points of $f(x)$?

Tip:

Always check for both vertical and horizontal asymptotes when analyzing rational functions—they provide key insights into the function's behavior!