Math Problem Statement
Graph the following piecewise function: f(x) = -2 for -6 < x < -3 and f(x) = x - 3 for -3 ≤ x < 2.
Solution
The function provided in your image is a piecewise function, defined as:
\begin{cases} -2 & \text{for } -6 < x < -3 \\ x - 3 & \text{for } -3 \leq x < 2 \end{cases}$$ Let's break this down step by step: ### 1. **First part**: $$f(x) = -2$$ for $$-6 < x < -3$$ - This represents a horizontal line where the $$y$$-value is always -2. - The line starts at $$x = -6$$ (not included, so open circle) and ends at $$x = -3$$ (also open circle). ### 2. **Second part**: $$f(x) = x - 3$$ for $$-3 \leq x < 2$$ - This is a linear equation with slope 1 and $$y$$-intercept -3. - The line begins at $$x = -3$$ with a closed circle (since it's $$\leq$$) and continues until $$x = 2$$ (open circle). ### Plotting Instructions: - Draw a horizontal line at $$y = -2$$ from $$x = -6$$ to $$x = -3$$, using open circles at both endpoints. - Plot the line $$f(x) = x - 3$$. Start at the point $$(-3, -6)$$ with a closed circle and end at $$(2, -1)$$ with an open circle. Would you like help plotting this graph or further clarification? Here are 5 questions for deeper understanding: 1. Why is there an open circle at $$x = -6$$ and $$x = 2$$? 2. How would the graph change if the function was $$f(x) = -2$$ for $$-6 \leq x \leq -3$$? 3. What is the slope of the line $$f(x) = x - 3$$ and how does it affect the graph? 4. How do you determine where the function is continuous or discontinuous? 5. Could you extend this function to other ranges of $$x$$? How? **Tip:** Always check the boundary conditions (open vs closed circles) to avoid mistakes when plotting piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Functions
Formulas
f(x) = -2 for -6 < x < -3
f(x) = x - 3 for -3 ≤ x < 2
Theorems
Graphing piecewise functions based on conditions
Open and closed circle notation in piecewise functions
Suitable Grade Level
Grades 9-11 (Algebra 2)