Math Problem Statement
Solution
To analyze this piecewise graph, we need to determine the domain and range of the function.
Domain:
The domain refers to all the x-values for which the function is defined. From the graph, it seems that the function is composed of three distinct segments:
- The first segment is a line increasing to the right, starting from approximately and stopping at with an open circle at , meaning the function is not defined at .
- The second segment is a parabola-like curve starting from and ending at . The function is not defined at (open circle), but it is defined at (closed circle).
- The third segment is a horizontal line starting at and continuing to the right, beyond the edge of the graph.
Thus, the domain is: It means that the function is not defined at , but it is defined for all other points in the intervals.
Range:
The range refers to all possible y-values of the function. Observing the graph:
- The first segment covers y-values from around to .
- The second segment (the parabola-like curve) reaches a minimum value at and a maximum at .
- The third segment is a horizontal line at .
Thus, the range is: This means that the function takes on all y-values between and 4 (including these values), and specifically, it takes due to the third segment.
Summary:
- Domain:
- Range:
Would you like to go into more detail or have any questions?
Related Questions:
- How do we formally define piecewise functions?
- What is the significance of open and closed circles in graphs?
- How does one interpret horizontal and vertical asymptotes in graphs?
- How do we algebraically find the domain and range of more complex functions?
- Can the range of a function ever be infinite in both directions?
Tip:
When determining the domain and range, always pay attention to open and closed circles, as they indicate whether the function includes or excludes those particular values at given points.
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Function Graphs
Formulas
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Theorems
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Suitable Grade Level
Grades 9-11