Math Problem Statement
find the graph
find the graph
Solution
The given piecewise function is defined as follows:
\begin{cases} -1 & \text{if } x \leq -2, \\ x + 1 & \text{if } -2 < x \leq 2, \\ 3 & \text{if } x > 4. \end{cases}$$ To graph this, we will: 1. **Graph the first part:** $$f(x) = -1$$ for $$x \leq -2$$. This is a horizontal line at $$y = -1$$, ending with a closed circle at $$x = -2$$. 2. **Graph the second part:** $$f(x) = x + 1$$ for $$-2 < x \leq 2$$. This is a straight line with slope 1 and y-intercept $$1$$, starting with an open circle at $$x = -2$$ and ending with a closed circle at $$x = 2$$. 3. **Graph the third part:** $$f(x) = 3$$ for $$x > 4$$. This is a horizontal line at $$y = 3$$, starting with an open circle at $$x = 4$$. Let me plot the graph for this function. numpy as np import matplotlib.pyplot as plt # Define the function values for the piecewise function x1 = np.linspace(-10, -2, 100) # For x <= -2 y1 = -1 * np.ones_like(x1) x2 = np.linspace(-2, 2, 100) # For -2 < x <= 2 y2 = x2 + 1 x3 = np.linspace(4, 10, 100) # For x > 4 y3 = 3 * np.ones_like(x3) # Plot the function plt.figure(figsize=(8, 6)) # Plot each segment plt.plot(x1, y1, label="f(x) = -1 for x ≤ -2", color="blue") plt.plot(x2, y2, label="f(x) = x + 1 for -2 < x ≤ 2", color="green") plt.plot(x3, y3, label="f(x) = 3 for x > 4", color="red") # Add markers for open and closed circles plt.plot(-2, -1, 'bo', label="Closed circle at x = -2 (f(x) = -1)", markersize=8) # Closed circle plt.plot(-2, -1, 'wo', markersize=6) # Open circle for clarity plt.plot(2, 3, 'go', label="Closed circle at x = 2 (f(x) = 3)", markersize=8) # Closed circle plt.plot(4, 3, 'ro', label="Open circle at x = 4 (f(x) = 3)", markersize=8) # Open circle # Labels and title plt.title("Graph of the Piecewise Function f(x)") plt.xlabel("x") plt.ylabel("f(x)") plt.axhline(0, color='black',linewidth=0.5) plt.axvline(0, color='black',linewidth=0.5) plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend() plt.show()Here is the graph of the given piecewise function $$f(x)$$. The blue segment represents $$f(x) = -1$$ for $$x \leq -2$$, the green segment represents $$f(x) = x + 1$$ for $$-2 < x \leq 2$$, and the red segment represents $$f(x) = 3$$ for $$x > 4$$. ### Key details: 1. A closed circle is used at the boundary where the value is included (e.g., $$x = -2$$ for $$f(x) = -1$$). 2. An open circle is used where the value is not included (e.g., $$x = 4$$ for $$f(x) = 3$$). Do you have any questions or need further clarification? --- Here are 5 related questions for further exploration: 1. What happens if the function is redefined for $$2 < x \leq 4$$? 2. How would the graph look if the condition $$x > 4$$ was changed to $$x \geq 4$$? 3. How can you find the range of the given piecewise function? 4. How would you calculate the domain if one segment was undefined for specific $$x$$-values? 5. How can you extend this piecewise function to include additional conditions for $$2 < x \leq 4$$? **Tip:** Always carefully consider open and closed circles in piecewise functions, as they indicate whether boundary values are included or excluded!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Linear Functions
Formulas
f(x) = -1 for x <= -2
f(x) = x + 1 for -2 < x <= 2
f(x) = 3 for x > 4
Theorems
Piecewise Function Theorem
Suitable Grade Level
Grades 9-11