Math Problem Statement
Solution
In this problem, you are asked to find the domain and range of the composition of functions , given the mappings of and between their respective domains and ranges.
Step 1: Analyze Function
From the left diagram:
- Domain of : {4, 5, 6, 8}
- Range of : {0, 1, 3, 5}
The mappings of are:
Step 2: Analyze Function
From the right diagram:
- Domain of : {0, 1, 3, 5} (which is the same as the range of )
- Range of : {2, 4, 6, 7}
The mappings of are:
Step 3: Composition
The composition means applying first, and then applying to the result. Let's go step by step for each element in the domain of :
Step 4: Find Domain and Range of
- Domain of : This is the domain of , which is {4, 5, 6, 8}.
- Range of : The outputs of the composition are {4, 2, 7, 6}.
Final Answer:
(a) Domain of : {4, 5, 6, 8}
(b) Range of : {2, 4, 6, 7}
Would you like any more details or clarifications on these concepts?
Here are 5 related questions for further practice:
- How do you find the composition of two functions algebraically, without mappings?
- What is the domain of a composition of functions if one function is undefined for certain values in its domain?
- Can the composition of two functions ever result in a function with a different domain than the original functions?
- How does the range of a function affect the composition with another function?
- Can functions be composed in the reverse order, i.e., ? How would the process differ?
Tip: Always check if the range of the first function matches the domain of the second function when composing two functions.
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Math Problem Analysis
Mathematical Concepts
Functions
Composition of Functions
Set Notation
Formulas
g ∘ f(x) = g(f(x))
Theorems
Composition of Functions Theorem
Suitable Grade Level
Grades 10-12