Math Problem Statement
If f(x) = sqrt(x^3 + 5), express f as the composition of two functions g(x) and h(x) such that g(h(x)) = f(x).
Solution
To express as the composition of two functions and , we need to identify two functions such that .
Solution:
Let:
- (this represents the "inside" function).
- (this represents the "outside" function).
Now, if we compose and , we get:
Thus, , and the decomposition is:
Let me know if you'd like more explanation or further help!
Here are 5 related questions for further exploration:
- What is the domain of ?
- How would and change if the function were ?
- How do you verify the composition step by step?
- Can you rewrite using another pair of and ?
- How does the chain rule relate to composing for derivative calculations?
Tip: When decomposing functions, focus on the outermost operation (e.g., square root, exponentiation) for , and let handle the inner expressions!
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Math Problem Analysis
Mathematical Concepts
Function Composition
Algebraic Functions
Square Roots
Formulas
f(x) = sqrt(x^3 + 5)
g(h(x)) = g(x^3 + 5) = sqrt(x^3 + 5)
Theorems
Function Composition Theorem
Suitable Grade Level
Grades 9-11