Unlike previous functions, rational functions require special handling when visualizing them in Python. The presence of undefined points and infinite behavior means we must carefully split the domain to avoid mathematical errors.

1. Defining the Function

We define our rational function as:

def rational_function(x):
    return 1 / (x - 1)

Key Considerations:

• $x = 1$ must be excluded from calculations to avoid division by zero;
• The function will be split into two domains (left and right of $x = 1$).

2. Splitting the Domain

To avoid division by zero, we generate two separate sets of x-values:

x_left = np.linspace(-4, 0.99, 250)  # Left of x = 1
x_right = np.linspace(1.01, 4, 250)  # Right of x = 1

The values 0.99 and 1.01 ensure we never include $x = 1$, preventing errors.

3. Plotting the Function

plt.plot(x_left, y_left, color='blue', linewidth=2, label=r"$f(x) = \frac{1}{x - 1}$")
plt.plot(x_right, y_right, color='blue', linewidth=2)

The function jumps at $x = 1$, so we need to plot it in two pieces.

4. Marking Asymptotes and Intercepts

• Vertical Asymptote ($x = 1$):

def rational_function(x):
    return 1 / (x - 1)

• Horizontal Asymptote ($y = 0$):

plt.axhline(0, color='green', linestyle='--', 
            linewidth=1, label="Horizontal Asymptote (y=0)")

• Y-Intercept at $x = 0$:

y_intercept = rational_function(0)
plt.scatter(0, y_intercept, color='purple', label="Y-Intercept")

5. Adding Directional Arrows

To indicate the function extends infinitely:

plt.annotate('', xy=(x_right[-1], y_right[-1]), xytext=(x_right[-2], y_right[-2]), arrowprops=dict(arrowstyle='->', color='blue', linewidth=1.5))