Math Problem Statement
Sketch a graph of f(x) = { 3 if x ≤ -2, -x + 1 if -2 < x ≤ 1, 1 if x > 1 }
Solution
The image you uploaded presents a piecewise function, defined as:
\begin{cases} 3 & \text{if } x \leq -2 \\ -x + 1 & \text{if } -2 < x \leq 1 \\ 1 & \text{if } x > 1 \end{cases}$$ Let's break down the function step-by-step and plot it on the provided graph: 1. **For $$x \leq -2$$:** The function is constant at $$f(x) = 3$$. This means there is a horizontal line at $$y = 3$$ for all $$x \leq -2$$. 2. **For $$-2 < x \leq 1$$:** The function is linear: $$f(x) = -x + 1$$, which has a slope of -1 and a y-intercept of 1. You can plot this as a straight line starting at $$(-2, 3)$$ and ending at $$(1, 0)$$. 3. **For $$x > 1$$:** The function is constant at $$f(x) = 1$$, which is a horizontal line at $$y = 1$$ for all $$x > 1$$. Now, let's describe what you should expect in the graph: - A **horizontal line at $$y = 3$$** extending leftward from $$x = -2$$ (including the point $$(-2, 3)$$). - A **linear segment** connecting $$(-2, 3)$$ to $$(1, 0)$$. - A **horizontal line at $$y = 1$$** starting from $$x = 1$$ and extending to the right. Would you like to know how to compute specific points or have more details? Here are 5 related questions to expand your understanding: 1. What is the significance of piecewise functions in real-world applications? 2. How do you find the slope and intercept of linear parts of a piecewise function? 3. How can we determine if the function is continuous at $$x = -2$$ and $$x = 1$$? 4. What is the domain and range of the piecewise function provided? 5. How does changing the slope of the linear segment affect the overall graph? **Tip:** Always mark open or closed circles at boundary points when graphing piecewise functions to show whether the endpoints are included or excluded.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Graphing
Formulas
y = -x + 1 for -2 < x ≤ 1
Theorems
Continuity of Piecewise Functions
Suitable Grade Level
Grades 10-12