Math Problem Statement
Decompose the function f, of, g, of, x, equals, cosine, x, to the power 5f(g(x))=cosx 5 into f, of, xf(x) and g, of, xg(x).
Solution
We are given the composite function . Our goal is to decompose this function into and .
To do this, we'll identify the inner function and the outer function .
Step 1: Identify
The function inside the cosine is , so we can define the inner function as:
Step 2: Identify
The outer function applies the cosine to the result of , which is . Since , the outer function must simply apply the cosine function. So:
Conclusion
Thus, the decomposition of is:
Would you like more details on function composition or anything else? Here are five related questions for deeper understanding:
- How would you decompose a function like ?
- What are the conditions for two functions to form a valid composite function?
- How do you find the derivative of a composite function using the chain rule?
- Can every function be decomposed into two functions? Why or why not?
- How would you graph the composite function ?
Tip: When decomposing a function, look for an "inner" function being acted upon by an "outer" function. The outer function typically operates on the result of the inner function.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Trigonometry
Formulas
f(g(x)) = f(x)
g(x) = x^5
f(x) = cos(x)
Theorems
Function Composition Theorem
Suitable Grade Level
Grades 10-12