The conditions given describe the behavior of the function $f(x)$ at specific points, which means we can sketch a graph that satisfies these criteria:

1. $f(2) = -1$: This indicates the function has the value $-1$ at $x = 2$, so the graph passes through the point $(2, -1)$.

2. $\lim_{x \to 2} f(x) = 2.5$: The limit of the function as $x$ approaches $2$ is $2.5$. This implies that the function's value at $x = 2$ is a discontinuity since $f(2) \neq 2.5$. The graph should have a jump discontinuity at $x = 2$.

3. $\lim_{x \to -3^+} f(x) = 0$: As $x$ approaches $-3$ from the right, the function approaches $0$. This indicates the function smoothly approaches the value $0$ from the right-hand side of $x = -3$.

4. $\lim_{x \to -3^-} f(x) = 2$: As $x$ approaches $-3$ from the left, the function approaches $2$. This implies a jump discontinuity at $x = -3$.

The graph will show two jump discontinuities: one at $x = 2$ and another at $x = -3$.

Let me create a plot illustrating this. matplotlib.pyplot as plt import numpy as np

Define points to visualize the conditions

x1 = np.linspace(-4, -3, 500) # Left of x = -3 x2 = np.linspace(-3, -2, 500) # Right of x = -3 x3 = np.linspace(1.5, 2, 500) # Left of x = 2 x4 = np.linspace(2, 2.5, 500) # Right of x = 2

Define the function behavior based on the conditions

y1 = 2 * np.ones_like(x1) # Approaching 2 from the left of -3 y2 = 0 * np.ones_like(x2) # Approaching 0 from the right of -3 y3 = 2.5 * np.ones_like(x3) # Approaching 2.5 from the left of 2 y4 = 2.5 * np.ones_like(x4) # Approaching 2.5 from the right of 2

Create the plot

plt.figure(figsize=(8, 6)) plt.plot(x1, y1, 'b-', label=r'$\lim_{{x \to -3^-}} f(x) = 2$') plt.plot(x2, y2, 'r-', label=r'$\lim_{{x \to -3^+}} f(x) = 0$') plt.plot(x3, y3, 'g-', label=r'$\lim_{{x \to 2}} f(x) = 2.5$ (Left of 2)') plt.plot(x4, y4, 'g-', label='nolegend') # Continue from the right of 2

Add the discrete points for f(2) = -1

plt.scatter([2], [-1], color='purple', zorder=5, label=r'$f(2) = -1$') plt.scatter([-3], [2], color='blue', zorder=5, label='Jump at x = -3 (Left Limit)') plt.scatter([-3], [0], color='red', zorder=5, label='Jump at x = -3 (Right Limit)')

Formatting the plot

plt.axvline(x=-3, color='k', linestyle='--', linewidth=0.7, label='x = -3') plt.axvline(x=2, color='k', linestyle='--', linewidth=0.7, label='x = 2') plt.axhline(0, color='black', linewidth=0.7) plt.title("Graph of the Function f(x)") plt.xlabel("x") plt.ylabel("f(x)") plt.legend() plt.grid() plt.show()The graph above represents a function $f(x)$ that satisfies all the given conditions. Key features include:

1. A jump discontinuity at $x = 2$ with $\lim_{x \to 2} f(x) = 2.5$ and $f(2) = -1$.
2. A jump discontinuity at $x = -3$ with $\lim_{x \to -3^-} f(x) = 2$ and $\lim_{x \to -3^+} f(x) = 0$.

Let me know if you need further details, or have other questions!

Here are 5 related questions for further exploration:

1. What is the formal definition of a jump discontinuity?
2. How do limits relate to the continuity of a function?
3. Can $f(x)$ have a point of continuity near $x = 2$ or $x = -3$?
4. What is the significance of $\lim_{x \to c} f(x)$ compared to $f(c)$?
5. How would this graph change if $f(2)$ were equal to $\lim_{x \to 2} f(x)$?

Tip: When dealing with discontinuities, always evaluate one-sided limits to fully understand the behavior of the function.