Math Problem Statement
Solution
The problem asks for the domain of the composition of two functions , using the graphs of (dashed line) and (solid line).
Step-by-step approach:
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Definition of Composition: The composition of two functions means that . For this to exist, the output of must lie in the domain of . Thus, we need to find where gives values that lie within the domain of .
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Domain of : From the graph, (the dashed line) is defined for since it only exists in this interval.
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Range of : (the solid line) is defined for , and the range of is from 0 to 3, inclusive.
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Condition for the Composition : For to exist, the output of must be within the domain of , which is from 0 to 3.
- takes values between 0 and 3 for .
Thus, the domain of the composition is the same as the domain of , which is .
Final Answer:
The domain of is .
Would you like further details or have any questions about this?
Here are 5 related questions for further exploration:
- What is the range of the composition function ?
- How does the composition behave graphically?
- How do we determine the domain of if the functions and are given algebraically?
- What is the importance of the range of in the composition ?
- How would the domain change if the graph of were different?
Tip: Always check the range of the inner function when determining the domain of a composition function.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Domain and Range
Formulas
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Theorems
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Suitable Grade Level
Grades 9-12