Math Problem Statement
Solution
To solve this problem, we need to find the domain and range of the composition of two functions , which is denoted as .
Step 1: Analyze the Domains and Ranges of and
From the diagrams provided:
- The domain of is the set of all possible input values for , which appears to be .
- The range of (which becomes the input for in the composition ) is .
For function :
- The domain of is , which matches the range of .
- The range of is .
Step 2: Determine the Domain and Range of
For the composition :
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Domain of : This will be the domain of , since must produce values in the range of that are within the domain of . Therefore, the domain of is .
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Range of : This will be the range of when applied to the range of . Observing the mappings, we see that:
This results in a range for of .
Final Answer
- (a) Domain of :
- (b) Range of :
Would you like further clarification on this solution or more details on the composition concept?
Related Questions:
- What is the general rule for determining the domain of a composite function?
- How can you identify the range of a function from a mapping diagram?
- What would happen if there were values in the range of that weren’t in the domain of ?
- How do you write compositions of functions in function notation?
- Can a composite function have a larger range than either of the original functions?
Tip:
When finding the range of a composition , always check each output of to see how it maps through . This will ensure you capture all possible outcomes for the composite function.
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Math Problem Analysis
Mathematical Concepts
Composition of Functions
Domain and Range
Formulas
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Theorems
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Suitable Grade Level
Grade 11-12