Math Problem Statement
Graph the following function on the axes provided. f, of, x, equals, left curly bracket, begin cases, begin case 1, x, plus, 2, for , x, is less than, minus, 3, end case 1, begin case 2, 1, for , x, equals, minus, 3, end case 2, begin case 3, 2, x, plus, 4, for , minus, 3, is less than, x, is less than or equal to, 1, end case 3, end cases, . f(x)= ⎩ ⎨ ⎧
x+2 1 2x+4
for x<−3 1 1
for x=−3 1 1
for −3<x≤1 1 1
Solution
Let's break down the piecewise function step by step and then graph it.
Function Definition:
The given function is defined as:
\begin{cases} x + 2 & \text{for } x < -3 \\ 1 & \text{for } x = -3 \\ 2x + 4 & \text{for } -3 < x \leq 1 \end{cases}$$ This means the function is defined differently for different ranges of $$x$$: 1. **For $$x < -3$$:** $$f(x) = x + 2$$ 2. **For $$x = -3$$:** $$f(x) = 1$$ 3. **For $$-3 < x \leq 1$$:** $$f(x) = 2x + 4$$ ### Graphing the Function: - **For $$x < -3$$:** The function $$f(x) = x + 2$$ is a straight line with a slope of 1 and a y-intercept of 2. This means as $$x$$ decreases, the function increases linearly. - **At $$x = -3$$:** The function value is exactly $$f(x) = 1$$, so there will be a discrete point on the graph at $$(-3, 1)$$. - **For $$-3 < x \leq 1$$:** The function $$f(x) = 2x + 4$$ is another straight line, with a slope of 2 and a y-intercept of 4. It starts from $$x = -3$$ and continues up to $$x = 1$$. --- I’ll go ahead and create a graph of this function to illustrate it. numpy as np import matplotlib.pyplot as plt # Define the function for each piece of the piecewise function def f1(x): return x + 2 # For x < -3 def f2(x): return 2*x + 4 # For -3 < x <= 1 # Generate x values for plotting x1 = np.linspace(-10, -3, 400) # For x < -3 x2 = np.linspace(-3, 1, 400) # For -3 < x <= 1 x3 = np.array([-3]) # Single point at x = -3 # Generate y values for each segment of the function y1 = f1(x1) y2 = f2(x2) y3 = np.array([1]) # Value of the function at x = -3 # Create the plot plt.figure(figsize=(8, 6)) plt.plot(x1, y1, label=r'$f(x) = x + 2$ for $x < -3$', color='blue') plt.plot(x2, y2, label=r'$f(x) = 2x + 4$ for $-3 < x \leq 1$', color='green') plt.scatter(x3, y3, color='red', zorder=5, label=r'$f(-3) = 1$') # Label the axes plt.axhline(0, color='black',linewidth=1) plt.axvline(0, color='black',linewidth=1) # Set the limits for x and y axis plt.xlim(-10, 2) plt.ylim(-5, 10) # Add title and labels plt.title('Graph of the Piecewise Function') plt.xlabel('x') plt.ylabel('f(x)') # Add grid and legend plt.grid(True) plt.legend() # Show the plot plt.show()Here is the graph of the piecewise function you provided: - The blue line represents the function $$f(x) = x + 2$$ for $$x < -3$$. - The red dot indicates the point $$f(-3) = 1$$, where the function takes the value 1 at $$x = -3$$. - The green line represents the function $$f(x) = 2x + 4$$ for $$-3 < x \leq 1$$. ### Would you like more details about this graph or any other aspect of the function? Feel free to ask if anything needs clarification! Here are some follow-up questions to deepen the understanding: 1. What is the slope of the function in the segment where $$x < -3$$? 2. How do the slopes of the two linear segments compare to each other? 3. How would the graph change if the function were defined differently at $$x = -3$$? 4. What is the y-intercept of the function for $$-3 < x \leq 1$$? 5. Can we extend the graph further? If so, how would it behave for $$x > 1$$? **Tip:** To analyze piecewise functions, it's often helpful to graph each piece separately and then combine them while carefully noting where the pieces meet or transition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing
Formulas
f(x) = x + 2 for x < -3
f(x) = 1 for x = -3
f(x) = 2x + 4 for -3 < x <= 1
Theorems
Piecewise Function Definition
Linear Function Slope-Intercept Form
Suitable Grade Level
Grades 9-11