Math Problem Statement
Graph the piecewise function g(x) defined as follows:
g(x) =
-1 if -1.5 ≤ x ≤ -0.5
0 if -0.5 ≤ x ≤ 0.5
1 if 0.5 < x ≤ 1.5
2 if 1.5 ≤ x ≤ 2.5
3 if 2.5 ≤ x < 3.5.
Solution
We are given a piecewise function with multiple intervals, and we need to graph it. Here’s a step-by-step breakdown of the function and how to graph it:
The function is defined as follows:
\begin{cases} -1 & \text{if } -1.5 \leq x \leq -0.5 \\ 0 & \text{if } -0.5 \leq x \leq 0.5 \\ 1 & \text{if } 0.5 < x \leq 1.5 \\ 2 & \text{if } 1.5 \leq x \leq 2.5 \\ 3 & \text{if } 2.5 \leq x < 3.5 \end{cases}$$ ### Step-by-step plotting: 1. **For $$-1.5 \leq x \leq -0.5$$, $$g(x) = -1$$**: - The function is a constant $$g(x) = -1$$ for all $$x$$ in this interval. - Draw a horizontal line at $$y = -1$$ from $$x = -1.5$$ to $$x = -0.5$$, inclusive of both endpoints. 2. **For $$-0.5 \leq x \leq 0.5$$, $$g(x) = 0$$**: - The function is a constant $$g(x) = 0$$ for all $$x$$ in this interval. - Draw a horizontal line at $$y = 0$$ from $$x = -0.5$$ to $$x = 0.5$$, inclusive of both endpoints. 3. **For $$0.5 < x \leq 1.5$$, $$g(x) = 1$$**: - The function is a constant $$g(x) = 1$$ for all $$x$$ in this interval. - Draw a horizontal line at $$y = 1$$ from $$x = 0.5$$ (not included, open circle) to $$x = 1.5$$ (included, closed circle). 4. **For $$1.5 \leq x \leq 2.5$$, $$g(x) = 2$$**: - The function is a constant $$g(x) = 2$$ for all $$x$$ in this interval. - Draw a horizontal line at $$y = 2$$ from $$x = 1.5$$ to $$x = 2.5$$, inclusive of both endpoints. 5. **For $$2.5 \leq x < 3.5$$, $$g(x) = 3$$**: - The function is a constant $$g(x) = 3$$ for all $$x$$ in this interval. - Draw a horizontal line at $$y = 3$$ from $$x = 2.5$$ (included, closed circle) to $$x = 3.5$$ (not included, open circle). ### Graph details: - For closed intervals (where $$x$$ is included), use a filled (closed) dot. - For open intervals (where $$x$$ is not included), use an open circle. Would you like me to provide any further details on this graph or a deeper explanation of the intervals? ### Follow-up questions: 1. How do we represent open and closed intervals on a graph? 2. Can you explain the difference between open and closed circles in graphing piecewise functions? 3. How do we handle discontinuities in a piecewise function when graphing? 4. What are the steps to identify where a function is continuous? 5. How do we interpret piecewise functions in real-world applications? **Tip:** When graphing piecewise functions, always ensure you properly mark the boundaries of each piece with either open or closed circles to reflect inclusivity or exclusivity of the points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Interval Notation
Constant Functions
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12