Math Problem Statement
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)equalsleft brace Start 2 By 3 Matrix 1st Row 1st Column x plus 2 2nd Column if 3rd Column x less than minus 3 2nd Row 1st Column x minus 2 2nd Column if 3rd Column x greater than or equals minus 3 EndMatrix
Question content area bottom Part 1 a. Choose the correct graph below. A. -10 10 -10 10 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 10 to 10 in increments of 2. A graph has two branches. The first branch is a ray that rises from right to left, starting at an open circle at (negative 3, negative 5) and passing through the points (negative 4, negative 4) and (negative 5, negative 3). The second branch is ray that falls from left to right, starting at a solid circle at (negative 3, negative 1) and passing through the points (negative 2, negative 2) and (negative 1, negative 3). B. -10 10 -10 10 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 10 to 10 in increments of 2. A graph has two branches. The first branch is a ray that falls from right to left, starting at a solid circle at (negative 3, 5) and passing through the points (negative 4, 4) and (negative 5, 3). The second branch is ray that rises from left to right, starting at an open circle at (negative 3, 1) and passing through the points (negative 2, 2) and (negative 1, 3). C. -10 10 -10 10 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 10 to 10 in increments of 2. A graph has two branches. The first branch is a ray that falls from right to left, starting at an open circle (negative 3, negative 1) and passing through the points (negative 9, negative 7) and (negative 10, negative 8). The second branch is ray that rises from left to right, starting at a solid circle at (negative 3, negative 5) and passing through the points (9, 7) and (10, 8). Your answer is correct.D. -10 10 -10 10 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 10 to 10 in increments of 2. A graph has two branches. The first branch is a ray that rises from right to left, starting at a solid circle at (negative 3, 1) and passing through the points (negative 9, 7) and (negative 10, 8). The second branch is ray that falls from left to right, starting at an open circle at (negative 3, 5) and passing through the points (9, negative 7) and (10, negative 8). 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 . (Use a comma to separate answers as needed.)
Solution
Part 1: Graph of the Piecewise Function
The function is given by the piecewise definition:
\begin{cases} x + 2 & \text{if } x < -3 \\ x - 2 & \text{if } x \geq -3 \end{cases}$$ To understand this, we can break the graph into two parts: 1. **For $$x < -3$$**: The function is $$f(x) = x + 2$$, which is a straight line with a slope of 1. Since $$x = -3$$ is not included in this branch, we will have an open circle at $$x = -3$$. 2. **For $$x \geq -3$$**: The function is $$f(x) = x - 2$$, which is another straight line with a slope of 1. This branch includes $$x = -3$$, so it will have a solid circle at $$x = -3$$. ### Let's match this with the given options: - **Option A**: This graph fits the description. It has an open circle at $$(-3, -5)$$ for the first branch, and a solid circle at $$(-3, -1)$$ for the second branch, indicating the correct behavior at $$x = -3$$. So, the correct answer for **Part 1** is **A**. --- ### Part 2: Range of the Piecewise Function Now, let's determine the **range** of the function based on its graph. 1. **For $$x < -3$$**: The line $$f(x) = x + 2$$ has a slope of 1, and since it starts from $$x = -3$$ with an open circle at $$f(-3) = -1$$, it increases without bound as $$x$$ decreases. Therefore, this branch's range is all values greater than $$-1$$. 2. **For $$x \geq -3$$**: The line $$f(x) = x - 2$$ also has a slope of 1, and since it starts at $$x = -3$$ with a solid circle at $$f(-3) = -5$$, it increases without bound as $$x$$ increases. This branch's range is all values greater than or equal to $$-5$$. Since both branches cover different parts of the range, we have the following: - From the first branch (for $$x < -3$$), we have values greater than $$-5$$. - From the second branch (for $$x \geq -3$$), we have values starting from $$-5$$ and increasing. Therefore, the **range** of the entire function is $$(-5, \infty)$$. ### The correct option for **Part 2** is: - **A**: The range does not have any isolated values. It can be described by $$(-5, \infty)$$. --- ### Summary of Answers: - **Part 1**: The correct graph is **A**. - **Part 2**: The range of the function is **$$(-5, \infty)$$**. --- Would you like more details or have any other questions on this? Here are some related questions: 1. How can you identify the type of graph for a piecewise function? 2. What does an open circle on a graph represent? 3. What does a solid circle on a graph represent? 4. How do you determine the domain and range of a piecewise function algebraically? 5. What are the key differences between linear and piecewise functions? **Tip**: When graphing piecewise functions, always check the behavior at the points where the function changes, and remember that open circles indicate values that are not included in that part of the function.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Functions
Domain and Range
Formulas
Piecewise function definition
Linear function equations (y = mx + b)
Theorems
Definition of piecewise functions
Concept of open and closed intervals in function graphs
Suitable Grade Level
Grades 9-12