Math Problem Statement
If the domain of f(x) is 1-4 (inclusive) and g(x) is 1-2 (inclusive) what is the domain of fg(x) and if gf(x)
Solution
To find the domains of and , we need to understand how function composition works with respect to domains.
1. For :
This is the composition of functions and , which means that we apply first and then apply to the result of . So, is only valid when:
- The output of lies within the domain of .
Given:
- The domain of is .
- The domain of is .
For to be valid, we need the output of to fall within . Since is defined for , we need to check if all outputs of (which lie between 1 and 2) are also within 's domain.
Thus, the domain of is the set of values of such that lies within . Since 's outputs are always in , and is defined for , the domain of is the domain of , which is .
2. For :
This is the composition , where we first apply and then apply to the result. For to be valid:
- The output of must lie within the domain of .
Given:
- The domain of is .
- The domain of is .
For to be valid, the output of must lie within . Since has a domain of , we need to find which values of produce outputs within . Therefore, we must check where .
The domain of is the set of values of such that . We do not have explicit details about the function , but we can infer that values for which lies in will give us the domain of . Thus, the domain of is a subset of , specifically where .
Summary:
- The domain of is .
- The domain of is a subset of , specifically where .
Would you like to explore further or clarify any points? Here are a few related questions:
- How does the range of a function affect its composition?
- Can you find the domain of a composition if the functions are given explicitly?
- What happens if the domain of one function does not overlap with the range of another in composition?
- How can function graphs help in visualizing composition domains?
- Can a function be composed with itself (i.e., )? What would the domain be?
Tip: Always remember to check that the range of the inner function fits within the domain of the outer function when composing them.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Domain of Functions
Formulas
Domain of (f ∘ g)(x) = Domain of g(x)
Domain of (g ∘ f)(x) = {x | f(x) ∈ Domain of g(x)}
Theorems
Composition of functions
Suitable Grade Level
Grades 9-11