Math Problem Statement
Given F(x) = cos²(x + 9), find functions f, g, and h such that F = f ∘ g ∘ h.
Solution
The given function is , and the problem asks to find functions , , and such that , i.e., .
To break down :
- We see that . This suggests we can work backwards by identifying the innermost function and gradually applying transformations.
Let's define the following functions:
- , which represents the shift inside the cosine function.
- , which applies the cosine operation to the result of .
- , which squares the result of .
So, the composition is:
Thus, the functions are:
Would you like more details on function compositions or have any questions? Here are some related questions to consider:
- What are the steps to decompose a function in terms of composition?
- How does the order of composition affect the final result?
- Can other function combinations also lead to the same result for ?
- How can we verify that our chosen functions are correct?
- What are some real-life applications of function composition?
Tip: When working with function compositions, always start by identifying the innermost transformation and work outward to match the overall expression.
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Math Problem Analysis
Mathematical Concepts
Function Composition
Trigonometry
Transformation of Functions
Formulas
F(x) = cos²(x + 9)
f(g(h(x))) = f( cos(h(x)) )
Theorems
Function Composition
Suitable Grade Level
Grades 11-12 (Pre-Calculus / Calculus Level)