Math Problem Statement
Solution
Let's go step by step with solving the problems visible in your image.
We are given the functions:
We need to compute the following:
1. Find :
This means applying inside , i.e., .
Substitute into :
Simplify:
2. Domain of :
The domain of depends on where the denominator is not zero. So, we solve:
Therefore, the domain of is .
3. Find :
This means applying inside , i.e., .
Substitute into :
To simplify:
4. Domain of :
The domain of depends on the domain of where 's denominator is not zero:
Thus, the domain of is .
5. Find :
This means applying inside , i.e., .
Substitute into itself:
Simplify:
6. Domain of :
The domain of is defined where the denominator is not zero:
Thus, the domain of is .
Would you like further clarification on any of these steps?
Here are 5 related questions:
- What happens when we swap the order of composition between two functions?
- How would you compute the inverse of or ?
- Can we verify the result of a composition by substituting specific values of ?
- What is the significance of domain restrictions in function compositions?
- How does the complexity of the function composition affect its graph?
Tip:
Always check the domain restrictions of each individual function before performing composition to avoid undefined operations.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Domain of Functions
Algebra
Formulas
(f ∘ g)(x) = f(g(x))
(g ∘ f)(x) = g(f(x))
Domain restrictions: Denominator ≠ 0
Theorems
Function Composition Theorem
Domain Restriction in Rational Functions
Suitable Grade Level
Grades 10-12
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