Math Problem Statement
The domain of the piecewise function is (minus−infinity∞,infinity∞). a. Graph the function. b. Use your graph to determine the function's range. f(x)equals=left 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 x+2 if x<−3 x−2 if x≥−3 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 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). 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 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). 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). 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 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). 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 EndSetenter your response here. (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 EndSetenter your response here. (Use a comma to separate answers as needed.)
Solution
Part 1: Graphing the Piecewise Function
The given piecewise function is:
\begin{cases} x + 2 & \text{if } x < -3 \\ x - 2 & \text{if } x \geq -3 \end{cases}$$ 1. For $$x < -3$$, the function is $$f(x) = x + 2$$. This is a linear equation with a slope of 1 and a y-intercept of 2. However, it is only valid for $$x < -3$$. Therefore, at $$x = -3$$, the function will have an open circle because $$x = -3$$ is not included in this part of the function. 2. For $$x \geq -3$$, the function is $$f(x) = x - 2$$. This is also a linear equation with a slope of 1 and a y-intercept of -2. It is valid for $$x \geq -3$$, meaning that the point at $$x = -3$$ will have a solid circle, as it includes the value $$x = -3$$. Let’s calculate key points: - For $$f(x) = x + 2$$ (valid when $$x < -3$$): - At $$x = -3$$, $$f(x) = -3 + 2 = -1$$ (but there is an open circle here). - At $$x = -5$$, $$f(x) = -5 + 2 = -3$$. - For $$f(x) = x - 2$$ (valid when $$x \geq -3$$): - At $$x = -3$$, $$f(x) = -3 - 2 = -5$$ (solid circle). - At $$x = -1$$, $$f(x) = -1 - 2 = -3$$. Based on this, the graph should have: - An open circle at $$(-3, -1)$$ for the first piece, and the line continuing leftwards. - A solid circle at $$(-3, -5)$$ for the second piece, with the line continuing to the right. **Correct Graph: Option B** - The graph in option B shows the open circle at $$(-3, -1)$$, a line rising to the left, and a solid circle at $$(-3, -5)$$, with the line rising to the right, which matches the description of the piecewise function. ### Part 2: Finding the Range Using the graph and the analysis of the function: - For $$x < -3$$, the function $$f(x) = x + 2$$ decreases without bound as $$x \to -\infty$$. Therefore, the range for this part is $$(-\infty, -1)$$. - For $$x \geq -3$$, the function $$f(x) = x - 2$$ starts at $$f(-3) = -5$$ (solid point) and increases without bound. Therefore, the range for this part is $$[-5, \infty)$$. Combining these two intervals, the range of the entire function is: $$(-\infty, -1) \cup [-5, \infty)$$ **Correct Answer for Part 2: Option B** - The range has at least one isolated value, and it can be described as the union of the intervals $$(-\infty, -1) \cup [-5, \infty)$$. ### Follow-up Questions: 1. How would the graph change if the condition for $$f(x)$$ switched at $$x = -2$$ instead of $$x = -3$$? 2. What happens to the graph if both pieces of the function are $$f(x) = x + 2$$ and $$f(x) = x - 2$$? 3. How can we find the intercepts of this piecewise function algebraically? 4. What are the differences in slopes for the two pieces of the function? 5. Can you describe how the range would change if the function was shifted vertically by 3 units? ### Tip: When graphing a piecewise function, pay attention to where the pieces change and whether the points at these transitions are open or closed circles.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing Linear Equations
Domain and Range
Formulas
f(x) = x + 2 (for x < -3)
f(x) = x - 2 (for x ≥ -3)
Theorems
Definition of Piecewise Functions
Concept of Intervals
Union of Sets
Suitable Grade Level
Grades 9-11