It looks like you want to graph the function:

$f(x) = \frac{x^3 + 1}{x^2 - 1}$

I'll generate the graph for you. numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return (x3 + 1) / (x2 - 1)

Define x values avoiding division by zero at x = ±1

x = np.linspace(-3, 3, 400) x = x[x != -1] # Remove x = -1 x = x[x != 1] # Remove x = 1

Compute y values

y = f(x)

Plot the function

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r'$f(x) = \frac{x^3 + 1}{x^2 - 1}$', color='b')

Mark vertical asymptotes at x = ±1

plt.axvline(x=-1, color='r', linestyle='dashed', label='Asymptote $x = -1$') plt.axvline(x=1, color='r', linestyle='dashed', label='Asymptote $x = 1$')

Add labels and title

plt.xlabel('x') plt.ylabel('f(x)') plt.title('Graph of $f(x) = \frac{x^3 + 1}{x^2 - 1}$') plt.legend() plt.grid()

Show the plot

plt.show()Here is the graph of $f(x) = \frac{x^3 + 1}{x^2 - 1}$. The function has vertical asymptotes at $x = \pm 1$, where the denominator equals zero. Let me know if you want any details!

Related Questions:

1. What are the horizontal and oblique asymptotes of this function?
2. How do I find the domain and range of $f(x)$?
3. What happens to $f(x)$ as $x \to \infty$ and $x \to -\infty$?
4. How can I determine the critical points and inflection points?
5. What are the intercepts of this function?

Tip:

Always check for asymptotes in rational functions by analyzing where the denominator equals zero!