Math Problem Statement
Question content area top Part 1 The domain of the piecewise function is (minusinfinity,infinity). a. Graph the function. b. Use your graph to determine the function's range. f(x)equalsStartSet Start 3 By 3 Matrix 1st Row 1st Column 0 2nd Column if 3rd Column x less than minus 3 2nd Row 1st Column negative 2 x 2nd Column if 3rd Column negative 3 less than or equals x less than 0 3rd Row 1st Column x squared 2nd Column if 3rd Column x greater than or equals 0 EndMatrix
Question content area bottom Part 1 a. Choose the correct graph below. A. -10 10 -20 20 x y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 20 to 20 in increments of 4. A graph has three branches. The first branch is a horizontal ray that extends to the left from an open circle at (negative 3, 0). The second branch is a line segment that falls from a solid circle at (negative 3, 6) to the point (0, 0). The third branch is a curve that falls from left to right at an increasing rate, starting at the point (0, 0) and passing through the points (1, negative 1) and (2, negative 4). B. -10 10 -20 20 x y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 20 to 20 in increments of 4. A graph has three branches. The first branch is a horizontal ray that extends to the left from an open circle at (negative 3, 0). The second branch is a line segment that falls from a solid circle at (negative 3, 6) to the point (0, 0). The third branch is a curve that rises from left to right at an increasing rate, starting at the point (0, 0) and passing through the points (1, 1) and (2, 4). C. -10 10 -20 20 x y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 20 to 20 in increments of 4. A graph has three branches. The first branch is a horizontal ray that extends to the left from an open circle at (negative 3, 0). The second branch is a line segment that rises from a solid circle at (negative 3, negative 6) to the point (0, 0). The third branch is a curve that falls from left to right at an increasing rate, starting at the point (0, 0) and passing through the points (1, negative 1) and (2, negative 4). D. -10 10 -20 20 x y
A coordinate system has a horizontal x-axis labeled from negative 10 to 10 in increments of 2 and a vertical y-axis labeled from negative 20 to 20 in increments of 4. A graph has three branches. The first branch is a curve that falls from left to right at an increasing rate passing through the points (negative 2, 4) and (negative 1, 1) to the point (0, 0). The second branch is a line segment that rises from the point (0, 0) to a solid circle at (3, 6). The third branch is a horizontal ray that extends to the right from an open circle at (3, 0). Part 2 b. What is the range of the entire piecewise function? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The range does not have any isolated values. It can be described by enter your response here. (Type your answer in interval notation.) B. The range has at least one isolated value. It can be described as the union of the interval(s) enter your response here and the set StartSet nothing EndSet . (Use a comma to separate answers as needed.) C. The range consists exclusively of one or more isolated values. It can be described as StartSet nothing EndSet
Solution
Part 1: Graph the Piecewise Function
The given piecewise function is:
\begin{cases} 0 & \text{if } x < -3 \\ -2x & \text{if } -3 \leq x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$$ Let's break it down and analyze the graph: 1. **For $$x < -3$$:** The function is a horizontal line at $$f(x) = 0$$, starting from $$x = -3$$ (open circle since $$x < -3$$) and extending to negative infinity. 2. **For $$-3 \leq x < 0$$:** The function is $$f(x) = -2x$$, which is a linear function with a slope of $$-2$$. It passes through the point $$(-3, 6)$$ (solid circle at $$-3$$ because it includes $$-3$$) and ends at the origin $$(0, 0)$$ (open circle since $$x < 0$$). 3. **For $$x \geq 0$$:** The function is $$f(x) = x^2$$, a parabola that opens upwards, starting from the origin $$(0, 0)$$ (solid circle at $$x = 0$$). Now, examining the graphs described: - **Graph A**: The third branch (curve) shows a parabola that falls, which is incorrect since $$x^2$$ is always non-negative and opens upwards. - **Graph B**: The third branch is a rising parabola from $$(0, 0)$$ passing through $$(1, 1)$$ and $$(2, 4)$$, which is correct for $$f(x) = x^2$$. Also, the second branch shows a line falling from $$(-3, 6)$$ to $$(0, 0)$$, which matches $$f(x) = -2x$$, and the first branch is a horizontal ray at $$f(x) = 0$$ for $$x < -3$$. - **Graph C**: The second branch rises from $$(-3, -6)$$, which is incorrect because $$f(x) = -2x$$ gives $$(-3, 6)$$, not $$(-3, -6)$$. - **Graph D**: The curve starts at negative values and is the wrong shape for $$x^2$$, and the linear section doesn't match the description. ### Conclusion: **The correct graph is Graph B.** --- ### Part 2: Determine the Range Using the graph, let's analyze the range of the function: 1. **For $$x < -3$$:** The function value is always $$0$$, so $$0$$ is an isolated value for this interval. 2. **For $$-3 \leq x < 0$$:** The function $$f(x) = -2x$$ decreases from $$f(-3) = 6$$ to $$f(0) = 0$$, covering all values between $$0$$ and $$6$$ (excluding $$0$$ since $$x = 0$$ is not included here). 3. **For $$x \geq 0$$:** The function $$f(x) = x^2$$ starts at $$f(0) = 0$$ and increases without bound, covering values from $$0$$ to infinity. ### Conclusion: The range includes: - $$0$$ as an isolated value from $$x < -3$$. - The interval $$(0, 6]$$ from the linear section. - The interval $$[0, \infty)$$ from the quadratic section. Thus, the range is: $$\boxed{[0, \infty)}$$ --- ### Follow-up Questions: 1. How do you graph piecewise functions with different expressions in each region? 2. Why is $$x^2$$ always non-negative for $$x \geq 0$$? 3. What are the key characteristics of linear and quadratic functions in graphs? 4. How do isolated values affect the range of a function? 5. Why is the origin a critical point for this piecewise function? ### Tip: When graphing piecewise functions, ensure each piece corresponds exactly to its defined domain, and check continuity or discontinuity at boundary points (like open or closed circles).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Functions
Quadratic Functions
Linear Functions
Interval Notation
Formulas
f(x) = 0 if x < -3
f(x) = -2x if -3 ≤ x < 0
f(x) = x^2 if x ≥ 0
Theorems
Range of a function
Properties of piecewise functions
Quadratic function behavior
Continuity and discontinuity at boundary points
Suitable Grade Level
Grades 10-12