You would like to graph the function:

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

Steps to Graph:

1. Factor the Denominator: Notice that the denominator $x^2 - 1$ is a difference of squares, so we can factor it:

   $x^2 - 1 = (x - 1)(x + 1)$

   This gives us the function:

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

2. Identify Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero (since the function will approach infinity). Set the denominator equal to zero:

   $(x - 1)(x + 1) = 0$

   This gives the points $x = 1$ and $x = -1$, so there are vertical asymptotes at $x = 1$ and $x = -1$.

3. Identify Horizontal Asymptote: As $x \to \infty$ or $x \to -\infty$, the degree of the numerator (which is 1) is less than the degree of the denominator (which is 2). Therefore, the horizontal asymptote is at $y = 0$.

4. Plot Key Points: You can substitute values of $x$ to find points on the graph. For example, at $x = 0$:

   $f(0) = \frac{0+2}{0^2 - 1} = \frac{2}{-1} = -2$

5. Sketch the Graph: The graph will have vertical asymptotes at $x = 1$ and $x = -1$, a horizontal asymptote at $y = 0$, and it will cross the y-axis at $(0, -2)$.

Let me plot this graph for you. numpy as np import matplotlib.pyplot as plt

Define the function

def f(x): return (x + 2) / (x**2 - 1)

Define the x range, avoiding division by zero at x = 1 and x = -1

x = np.linspace(-5, 5, 400) x = x[np.abs(x - 1) > 0.05] # Avoid x = 1 x = x[np.abs(x + 1) > 0.05] # Avoid x = -1

Calculate y values

y = f(x)

Plot the graph

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

Plot vertical asymptotes at x = 1 and x = -1

plt.axvline(x=1, color='r', linestyle='--', label='Vertical Asymptote at x = 1') plt.axvline(x=-1, color='g', linestyle='--', label='Vertical Asymptote at x = -1')

Plot horizontal asymptote at y = 0

plt.axhline(y=0, color='k', linestyle='--', label='Horizontal Asymptote at y = 0')

Mark the y-intercept at (0, -2)

plt.scatter(0, -2, color='purple', zorder=5) plt.text(0, -2, ' (0, -2)', color='purple', verticalalignment='bottom')

Set labels and title

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

Set the limits for better viewing

plt.xlim([-5, 5]) plt.ylim([-10, 10])

Show the plot

plt.grid(True) plt.show()Here is the graph of the function $f(x) = \frac{x+2}{x^2 - 1}$. The vertical asymptotes are located at $x = 1$ and $x = -1$, and the horizontal asymptote is at $y = 0$. The graph also shows that the function passes through the point $(0, -2)$.

Would you like more details or clarification on any part of the graph?

Here are some related questions to further explore the function:

1. How do you determine the behavior of the function near the vertical asymptotes?
2. What happens to $f(x)$ as $x \to \infty$?
3. Can we identify any intervals where the function is increasing or decreasing?
4. How would the graph change if we modified the numerator of the function?
5. What are the general steps to graph a rational function like this?

Tip: Always check for vertical asymptotes by setting the denominator equal to zero and solving for the values of $x$. This helps determine where the function is undefined and where it may approach infinity.