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 f(g(x))=cos(x5)f(g(x)) = \cos(x^5). Our goal is to decompose this function into f(x)f(x) and g(x)g(x).

To do this, we'll identify the inner function g(x)g(x) and the outer function f(x)f(x).

Step 1: Identify g(x)g(x)

The function inside the cosine is x5x^5, so we can define the inner function as: g(x)=x5g(x) = x^5

Step 2: Identify f(x)f(x)

The outer function applies the cosine to the result of g(x)g(x), which is cos(g(x))\cos(g(x)). Since g(x)=x5g(x) = x^5, the outer function must simply apply the cosine function. So: f(x)=cos(x)f(x) = \cos(x)

Conclusion

Thus, the decomposition of f(g(x))=cos(x5)f(g(x)) = \cos(x^5) is: g(x)=x5g(x) = x^5 f(x)=cos(x)f(x) = \cos(x)

Would you like more details on function composition or anything else? Here are five related questions for deeper understanding:

  1. How would you decompose a function like f(g(x))=sin(2x+3)f(g(x)) = \sin(2x + 3)?
  2. What are the conditions for two functions to form a valid composite function?
  3. How do you find the derivative of a composite function using the chain rule?
  4. Can every function be decomposed into two functions? Why or why not?
  5. How would you graph the composite function f(g(x))=cos(x5)f(g(x)) = \cos(x^5)?

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