Math Problem Statement
Graph the following function on the axes provided.
f, of, x, equals, left curly bracket, begin cases, begin case 1, minus, 2, for , minus, 6, is less than, x, is less than, minus, 3, end case 1, begin case 2, x, minus, 3, for , minus, 3, is less than or equal to, x, is less than, 2, end case 2, end cases, .
f(x)={−2x−3 for −6<x<−311 for −3≤x<211
LineClosed CircleOpen Circle
Solution
To graph the piecewise function on the specified interval, let's analyze each piece:
The function is defined as:
\begin{cases} -2 & \text{for } -6 < x < -3 \\ x - 3 & \text{for } -3 \leq x < 2 \end{cases}$$ ### Step 1: Plot the First Case 1. **Function**: $$f(x) = -2$$ 2. **Interval**: $$-6 < x < -3$$ 3. **Behavior**: The function is constant at $$f(x) = -2$$ for all $$x$$ in this interval. - **Open Circle** at $$x = -6$$: Since $$x$$ does not equal $$-6$$, the point $$(-6, -2)$$ should be represented with an open circle. - **Open Circle** at $$x = -3$$: Similarly, $$(-3, -2)$$ should also be represented with an open circle. ### Step 2: Plot the Second Case 1. **Function**: $$f(x) = x - 3$$ 2. **Interval**: $$-3 \leq x < 2$$ 3. **Behavior**: This part of the function is a linear equation with a slope of 1. - **Closed Circle** at $$x = -3$$: Since $$x = -3$$ is included in this interval, $$(-3, -6)$$ is a closed circle point. - **Open Circle** at $$x = 2$$: Since $$x = 2$$ is not included in this interval, $$(2, -1)$$ is an open circle point. ### Plotting Points for the Graph Based on this analysis: 1. For $$-6 < x < -3$$, draw a horizontal line at $$y = -2$$ with open circles at $$(-6, -2)$$ and $$(-3, -2)$$. 2. For $$-3 \leq x < 2$$, draw a line with points from $$(-3, -6)$$ (closed circle) to $$(2, -1)$$ (open circle). Would you like to see the plotted graph, or do you have any questions? --- Here are five related questions: 1. How would the graph change if the interval for the second case was modified? 2. What would happen if the inequality signs were all changed to strict inequalities? 3. How do open and closed circles affect the continuity of a piecewise function? 4. How do you determine where to place open and closed circles on a piecewise graph? 5. How would the graph look if a third case were added to the piecewise function? **Tip:** When graphing piecewise functions, always start by identifying the domain of each piece and plotting each segment separately.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Inequalities
Graphing Functions
Formulas
f(x) = { -2 for -6 < x < -3, x - 3 for -3 ≤ x < 2 }
Theorems
Piecewise Function Graphing
Suitable Grade Level
Grades 9-11