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 f(x)=x3+5f(x) = \sqrt{x^3 + 5} as the composition of two functions g(x)g(x) and h(x)h(x), we need to identify two functions such that g(h(x))=f(x)g(h(x)) = f(x).

Solution:

Let:

  1. h(x)=x3+5h(x) = x^3 + 5 (this represents the "inside" function).
  2. g(x)=xg(x) = \sqrt{x} (this represents the "outside" function).

Now, if we compose gg and hh, we get: g(h(x))=g(x3+5)=x3+5.g(h(x)) = g(x^3 + 5) = \sqrt{x^3 + 5}.

Thus, g(h(x))=f(x)g(h(x)) = f(x), and the decomposition is: h(x)=x3+5,g(x)=x.h(x) = x^3 + 5, \quad g(x) = \sqrt{x}.

Let me know if you'd like more explanation or further help!


Here are 5 related questions for further exploration:

  1. What is the domain of f(x)=x3+5f(x) = \sqrt{x^3 + 5}?
  2. How would g(x)g(x) and h(x)h(x) change if the function were f(x)=x35f(x) = \sqrt{x^3 - 5}?
  3. How do you verify the composition g(h(x))=f(x)g(h(x)) = f(x) step by step?
  4. Can you rewrite f(x)=x3+5f(x) = \sqrt{x^3 + 5} using another pair of g(x)g(x) and h(x)h(x)?
  5. How does the chain rule relate to composing g(h(x))g(h(x)) for derivative calculations?

Tip: When decomposing functions, focus on the outermost operation (e.g., square root, exponentiation) for g(x)g(x), and let h(x)h(x) 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