Yes! I can generate a sketch of this graph for you. I'll plot the different segments, mark the discontinuities, and ensure the transitions are clear. Let me create the graph now. numpy as np import matplotlib.pyplot as plt

Define piecewise function domains

x1 = np.linspace(-6, -3, 100, endpoint=False) # For x < -3 x2 = np.linspace(-3, 3, 100, endpoint=False) # For -3 ≤ x < 3 x3 = np.linspace(3, 5, 100, endpoint=False) # For 3 < x ≤ 5 x4 = np.linspace(5.1, 8, 100) # For x > 5 (excluding 5 to avoid ln(0))

Define functions

y1 = -x1 - 5 y2 = x2 y3 = 2*x3 - 3 y4 = np.log(x4 - 5)

Plot the functions

plt.figure(figsize=(8, 6))

plt.plot(x1, y1, 'b', label=r'$f(x)=-x-5, \quad x<-3$') plt.plot(x2, y2, 'g', label=r'$f(x)=x, \quad -3\leq x<3$') plt.plot(x3, y3, 'r', label=r'$f(x)=2x-3, \quad 3<x\leq5$') plt.plot(x4, y4, 'm', label=r'$f(x)=\ln(x-5), \quad x>5$')

Mark points of interest

plt.scatter([-3, 3, 5], [-(-3)-5, 3, 2*5-3], color='black', zorder=3) # Filled points plt.scatter([-3, 5], [(-3), np.log(5-5)], color='white', edgecolors='black', zorder=3) # Hollow points

Labels and title

plt.axvline(x=-3, color='gray', linestyle='--', alpha=0.5) # Vertical guide at x = -3 plt.axvline(x=3, color='gray', linestyle='--', alpha=0.5) # Vertical guide at x = 3 plt.axvline(x=5, color='gray', linestyle='--', alpha=0.5) # Vertical guide at x = 5 plt.axhline(y=0, color='black', linewidth=0.5) plt.axvline(x=0, color='black', linewidth=0.5)

plt.xlabel('x') plt.ylabel('f(x)') plt.title('Piecewise Function with Discontinuities') plt.legend() plt.grid(True, linestyle='--', alpha=0.6)

Show the plot

plt.show()Here's the graph of the given piecewise function.

• A jump discontinuity is present at $x = -3$ (the function jumps from one value to another).
• The function is continuous at $x = 3$ (smooth transition).
• An infinite discontinuity is present at $x = 5$ because $\ln(0)$ is undefined.

Let me know if you have any questions or if you'd like any modifications!

Related Questions:

1. How do you determine whether a function is continuous at a given point?
2. What are the different types of discontinuities in a function?
3. How does the derivative behave at points of discontinuity?
4. Can you find the left-hand and right-hand limits at $x = -3$ and $x = 5$?
5. How would you modify the function to remove the discontinuity at $x = -3$?

Tip:

When analyzing piecewise functions, always check for continuity by comparing left-hand and right-hand limits at transition points!