Math Problem Statement

Decompose the function f, of, g, of, x, equals, x, to the power 9 , minus, 9f(g(x))=x 9 −9 into f, of, xf(x) and g, of, xg(x). Answer Attempt 1 out of 2

g, of, x, equalsg(x)= f, of, x, equalsf(x)=

Solution

To decompose the function f(g(x))=x99f(g(x)) = x^9 - 9, we aim to express the function in terms of two simpler functions, f(x)f(x) and g(x)g(x), such that:

f(g(x))=x99f(g(x)) = x^9 - 9

This means we want to find functions f(x)f(x) and g(x)g(x) where:

f(g(x))=f(something)=something simplerf(g(x)) = f(\text{something}) = \text{something simpler}

Step 1: Guess the form of g(x)g(x)

Let's assume that the simpler function is a power of xx. We start by setting:

g(x)=x9g(x) = x^9

Step 2: Find f(x)f(x)

Now that we have g(x)=x9g(x) = x^9, we substitute this into the original equation to find f(x)f(x). From the equation:

f(g(x))=x99f(g(x)) = x^9 - 9

Substituting g(x)=x9g(x) = x^9 into this:

f(x9)=x99f(x^9) = x^9 - 9

This implies that:

f(x)=x9f(x) = x - 9

Final Decomposition

  • g(x)=x9g(x) = x^9
  • f(x)=x9f(x) = x - 9

Would you like a more detailed explanation or have any questions?

Related Questions

  1. How do we verify the decomposition of functions?
  2. What are other possible forms for g(x)g(x) and f(x)f(x)?
  3. How would we graph f(g(x))f(g(x)) to check correctness?
  4. What is the general approach to decomposing functions?
  5. How do compositions of functions behave with respect to operations like addition or multiplication?

Tip:

When decomposing a function, it helps to think of the inner function as handling the most complex part, leaving the outer function to handle the remainder.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Functions
Decomposition of Functions

Formulas

f(g(x)) = x^9 - 9
g(x) = x^9
f(x) = x - 9

Theorems

Function Decomposition

Suitable Grade Level

Grades 10-12