Math Problem Statement
Graph the piecewise function and determine its range from the graph. The function is defined as: f(x) = { 5 if x ≤ 2, -5 if x > 2 }.
Solution
The piecewise function given is:
5 & \text{if } x \leq 2 \\ -5 & \text{if } x > 2 \end{cases}$$ ### Part a. Graphing the function: - For $$x \leq 2$$, the function is a constant $$f(x) = 5$$. This represents a horizontal line at $$y = 5$$ for all values of $$x$$ less than or equal to 2. - For $$x > 2$$, the function is a constant $$f(x) = -5$$. This represents a horizontal line at $$y = -5$$ for all values of $$x$$ greater than 2. At $$x = 2$$, the value of the function is $$5$$, so there will be a closed circle (indicating inclusion) at $$(2, 5)$$ and an open circle (indicating exclusion) at $$(2, -5)$$ since the function jumps to $$-5$$ after $$x = 2$$. ### Correct graph: The correct graph should have: - A horizontal line at $$y = 5$$ from $$x = -\infty$$ to $$x = 2$$, with a closed dot at $$(2, 5)$$. - A horizontal line at $$y = -5$$ starting at $$x = 2$$ but with an open circle at $$(2, -5)$$. **Answer:** The correct graph is **option D**. This graph satisfies both conditions with the closed circle at $$(2, 5)$$ and open circle at $$(2, -5)$$. ### Part b. Range of the function: The range of the function is the set of all possible $$y$$-values that the function can take. From the graph, we observe that the function takes only two values: $$5$$ and $$-5$$. Thus, the range of the function is $$\{ 5, -5 \}$$. --- Would you like a deeper explanation of any part, or have further questions? Here are 5 related questions you might find useful: 1. What does a piecewise function represent in general? 2. How would the graph change if the conditions on $$x$$ were different? 3. Can you explain what open and closed circles mean on a graph? 4. What other types of piecewise functions can you graph using similar steps? 5. How can the range of a piecewise function change depending on its conditions? **Tip:** Always pay attention to whether a point on the graph includes or excludes certain values by checking for open (unfilled) or closed (filled) circles at the boundaries.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Graphing
Range of a Function
Formulas
Piecewise Function Definition
Range Determination from a Graph
Theorems
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Suitable Grade Level
Grades 8-10