Math Problem Statement
Given the one-to-one function f(x) = ∛(1 - x), find f^{-1}(x).
Solution
We are given the one-to-one function , and we need to find its inverse, .
To find the inverse of a function, we follow these steps:
Steps to find :
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Rewrite the function with :
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Swap and :
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Solve for :
- Cube both sides to eliminate the cube root:
- Solve for :
Thus, the inverse function is:
Final Answer:
Would you like further clarification or additional details on any of the steps?
Here are some related questions to explore further:
- What is the domain and range of and its inverse?
- How would you verify that and are indeed inverses?
- How does the graph of relate to the graph of ?
- What are the general conditions for a function to have an inverse?
- How does the cubic root function affect the shape of the graph?
Tip: When finding the inverse of a function, always make sure that the original function is one-to-one (injective), which guarantees that its inverse exists.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Cube Roots
Algebra
Formulas
f(x) = ∛(1 - x)
f^{-1}(x) = 1 - x^3
Theorems
Inverse function theorem
Suitable Grade Level
Grades 10-12