Math Problem Statement

Suppose H(x) = 4x^5 - 3. Find two functions f and g such that (f ∘ g)(x) = H(x). Neither function can be the identity function.

Solution

The problem asks to find two functions, f(x)f(x) and g(x)g(x), such that their composition f(g(x))=H(x)f(g(x)) = H(x), where H(x)=4x53H(x) = 4x^5 - 3. The additional condition is that neither function can be the identity function.

To solve this, we need to decompose the function H(x)=4x53H(x) = 4x^5 - 3 into a composition of two simpler functions.

Decomposition Approach:

One possible decomposition is:

  • Let g(x)=x5g(x) = x^5, which simplifies part of the expression.
  • Then, let f(x)=4x3f(x) = 4x - 3.

Now let's check this:

  • If g(x)=x5g(x) = x^5, then applying f(x)f(x) to g(x)g(x) gives: f(g(x))=f(x5)=4(x5)3=4x53.f(g(x)) = f(x^5) = 4(x^5) - 3 = 4x^5 - 3.

Thus, f(g(x))=H(x)f(g(x)) = H(x), as required.

Final Answer:

  • f(x)=4x3f(x) = 4x - 3
  • g(x)=x5g(x) = x^5

This is one possible solution, but there may be other valid decompositions as well.

Would you like further details or have any questions? Here are some related questions you could explore:

  1. Can we decompose H(x)=4x53H(x) = 4x^5 - 3 in another way?
  2. How do you verify that f(g(x))=H(x)f(g(x)) = H(x) for any pair of functions?
  3. What is the significance of the identity function in functional decomposition?
  4. How would the decomposition change if the function H(x)H(x) had different powers of xx?
  5. Can compositions like these be applied to functions that involve transcendental elements like exponentials or logarithms?

Tip: When composing functions, always evaluate the inner function first, then apply the outer function to its result.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra

Formulas

(f ∘ g)(x) = f(g(x))

Theorems

Function Decomposition

Suitable Grade Level

Grades 9-12