Math Problem Statement
Graph the piecewise function defined as follows: g(x) = -3 if -1.5 < x ≤ -0.5, -2 if -0.5 < x ≤ 0.5, -1 if 0.5 < x < 1.5, 0 if 1.5 ≤ x < 2.5, 1 if 2.5 ≤ x < 3.5.
Solution
The given function is a piecewise function defined as follows:
\begin{cases} -3 & \text{if } -1.5 < x \leq -0.5 \\ -2 & \text{if } -0.5 < x \leq 0.5 \\ -1 & \text{if } 0.5 < x < 1.5 \\ 0 & \text{if } 1.5 \leq x < 2.5 \\ 1 & \text{if } 2.5 \leq x < 3.5 \end{cases}$$ ### Steps to Graph the Function: 1. **Segment 1:** From $$x = -1.5$$ to $$x = -0.5$$, $$g(x) = -3$$. Mark this as a horizontal line at $$y = -3$$ from $$x = -1.5$$ (open circle) to $$x = -0.5$$ (closed circle). 2. **Segment 2:** From $$x = -0.5$$ to $$x = 0.5$$, $$g(x) = -2$$. Mark a horizontal line at $$y = -2$$ from $$x = -0.5$$ (open circle) to $$x = 0.5$$ (closed circle). 3. **Segment 3:** From $$x = 0.5$$ to $$x = 1.5$$, $$g(x) = -1$$. Mark a horizontal line at $$y = -1$$ from $$x = 0.5$$ (open circle) to $$x = 1.5$$ (closed circle). 4. **Segment 4:** From $$x = 1.5$$ to $$x = 2.5$$, $$g(x) = 0$$. Mark a horizontal line at $$y = 0$$ from $$x = 1.5$$ (closed circle) to $$x = 2.5$$ (open circle). 5. **Segment 5:** From $$x = 2.5$$ to $$x = 3.5$$, $$g(x) = 1$$. Mark a horizontal line at $$y = 1$$ from $$x = 2.5$$ (closed circle) to $$x = 3.5$$ (closed circle). Would you like further details on how to graph this or a deeper explanation of any part? --- ### Related Questions: 1. What are the key differences between open and closed circles in graphing piecewise functions? 2. How does the continuity of $$g(x)$$ vary across different intervals? 3. Can we determine limits at points where the function jumps, like at $$x = -0.5$$? 4. How would we describe $$g(x)$$ in terms of discontinuities? 5. What real-world scenarios might model a piecewise function like $$g(x)$$? #### Tip: Always check for open and closed circles carefully to ensure you mark the endpoints correctly when graphing piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Algebra
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12