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 , we aim to express the function in terms of two simpler functions, and , such that:
This means we want to find functions and where:
Step 1: Guess the form of
Let's assume that the simpler function is a power of . We start by setting:
Step 2: Find
Now that we have , we substitute this into the original equation to find . From the equation:
Substituting into this:
This implies that:
Final Decomposition
Would you like a more detailed explanation or have any questions?
Related Questions
- How do we verify the decomposition of functions?
- What are other possible forms for and ?
- How would we graph to check correctness?
- What is the general approach to decomposing functions?
- 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