Math Problem Statement
GRAPH THE FUNCTION
Solution
To graph the function defined by the following piecewise function:
\begin{cases} 3 & \text{if } x < 0 \\ -3 & \text{if } x = 0 \\ 2 & \text{if } x > 0 \end{cases}$$ We need to follow these steps: 1. **Draw the horizontal lines for each condition**: - For $$x < 0$$: Draw a horizontal line at $$y = 3$$. This line should continue infinitely to the left but should stop just before reaching $$x = 0$$. - For $$x > 0$$: Draw a horizontal line at $$y = 2$$. This line should continue infinitely to the right but should stop just after $$x = 0$$. 2. **Plot the point for $$x = 0$$**: - At $$x = 0$$, the function value is $$-3$$. So, plot a point at $$(0, -3)$$. This point should be a filled-in circle, indicating that the function equals $$-3$$ exactly at $$x = 0$$. ### Graph Construction: - **Left of the origin**: Draw a solid line parallel to the x-axis at $$y = 3$$ extending leftward from $$x = 0$$ without including $$x = 0$$. Use an open circle (unfilled dot) at $$(0, 3)$$ to show that the function does not include this value at $$x = 0$$. - **At the origin**: Place a filled dot at $$(0, -3)$$. - **Right of the origin**: Draw a solid line parallel to the x-axis at $$y = 2$$ extending rightward from $$x = 0$$ without including $$x = 0$$. Use an open circle (unfilled dot) at $$(0, 2)$$ to indicate that this part does not include the point at $$x = 0$$. ### Final Notes: - Ensure the horizontal lines are distinctly separated at $$y = 3$$ for $$x < 0$$ and $$y = 2$$ for $$x > 0$$. - Make sure the point at $$x = 0$$ is clearly marked at $$-3$$. Would you like to see a sketch of this graph, or do you have any further questions? --- Here are 5 related questions you might find useful: 1. What is a piecewise function, and how is it defined? 2. How do you determine the continuity of a piecewise function? 3. What is the difference between open and closed circles when graphing piecewise functions? 4. How can you determine the domain and range of a piecewise function? 5. How would the graph change if one of the piecewise conditions had a different inequality, like $$x \leq 0$$? **Tip:** When graphing piecewise functions, always carefully observe the conditions and include or exclude endpoints as specified.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Graphing functions
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12