Math Problem Statement
Graph the following piecewise functions:
(a) f(x) = {3x - 1, if x <= 0; -2x + 1, if x > 0}
(b) f(x) = {-2x - 1, if x < -1; 1, if -1 <= x <= 1; 2x - 1, if x > 1}
Solution
The task requires graphing two piecewise functions.
Let’s break down each function.
a)
The given piecewise function is:
\begin{cases} 3x - 1 & \text{if } x \leq 0 \\ -2x + 1 & \text{if } x > 0 \end{cases}$$ - For $$x \leq 0$$, the equation is $$f(x) = 3x - 1$$, which is a straight line. - For $$x > 0$$, the equation is $$f(x) = -2x + 1$$, which is another straight line. ### b) $$f(x)$$ The second function is: $$f(x) = \begin{cases} -2x - 1 & \text{if } x < -1 \\ 1 & \text{if } -1 \leq x \leq 1 \\ 2x - 1 & \text{if } x > 1 \end{cases}$$ - For $$x < -1$$, the equation is $$f(x) = -2x - 1$$, which is a line with a negative slope. - For $$-1 \leq x \leq 1$$, the value of the function is constant and equal to 1. - For $$x > 1$$, the equation is $$f(x) = 2x - 1$$, which is a line with a positive slope. I will now generate the graphs for both piecewise functions. numpy as np import matplotlib.pyplot as plt # Define the function a(x) based on the given piecewise function def f_a(x): return np.piecewise(x, [x <= 0, x > 0], [lambda x: 3*x - 1, lambda x: -2*x + 1]) # Define the function b(x) based on the given piecewise function def f_b(x): return np.piecewise(x, [x < -1, (-1 <= x) & (x <= 1), x > 1], [lambda x: -2*x - 1, 1, lambda x: 2*x - 1]) # Create an array of x values for plotting x_a = np.linspace(-5, 5, 400) x_b = np.linspace(-5, 5, 400) # Calculate the corresponding y values y_a = f_a(x_a) y_b = f_b(x_b) # Plot the graphs plt.figure(figsize=(12, 6)) # Plot for function a plt.subplot(1, 2, 1) plt.plot(x_a, y_a, label="f(x) = 3x - 1 (x <= 0), f(x) = -2x + 1 (x > 0)") plt.axvline(0, color='black', linestyle='--', linewidth=1) plt.axhline(0, color='black', linestyle='--', linewidth=1) plt.title('Graph of f(x) for function a') plt.xlabel('x') plt.ylabel('f(x)') plt.grid(True) plt.legend() # Plot for function b plt.subplot(1, 2, 2) plt.plot(x_b, y_b, label="f(x) = -2x - 1 (x < -1), f(x) = 1 (-1 <= x <= 1), f(x) = 2x - 1 (x > 1)") plt.axvline(-1, color='black', linestyle='--', linewidth=1) plt.axvline(1, color='black', linestyle='--', linewidth=1) plt.axhline(0, color='black', linestyle='--', linewidth=1) plt.title('Graph of f(x) for function b') plt.xlabel('x') plt.ylabel('f(x)') plt.grid(True) plt.legend() # Show the plots plt.tight_layout() plt.show()Here are the graphs for both functions: ### a) Function $$f(x)$$ - For $$x \leq 0$$, the function is represented by the line $$f(x) = 3x - 1$$. - For $$x > 0$$, the function is represented by the line $$f(x) = -2x + 1$$. ### b) Function $$f(x)$$ - For $$x < -1$$, the function is represented by the line $$f(x) = -2x - 1$$. - For $$-1 \leq x \leq 1$$, the function is constant, $$f(x) = 1$$. - For $$x > 1$$, the function is represented by the line $$f(x) = 2x - 1$$. Let me know if you would like any further explanations or details on how these graphs were constructed. Also, here are 5 questions for further study: 1. How does the slope of a line affect its graph, particularly in piecewise functions? 2. How can we calculate the values of these functions for specific $$x$$ values? 3. What is the significance of discontinuities in piecewise functions? 4. How do we find the intersection points of two piecewise functions? 5. How can we apply piecewise functions to real-life situations, like economics or physics? **Tip**: When graphing piecewise functions, always pay attention to the boundaries where the function changes its form (like $$x = 0$$ or $$x = -1$$, etc.), as these are crucial for accurately plotting the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Linear Equations
Formulas
f(x) = mx + b for linear equations
Piecewise representation for functions
Theorems
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Suitable Grade Level
Grades 9-11