Math Problem Statement
The one-to-one function is defined below. f(x)= ^3√3-x+7 Find f^-1(x) where f^-1 is the inverse of f.
Solution
To find the inverse function , we need to reverse the process that the function applies to its input. The general steps are as follows:
Steps to Find the Inverse of :
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Rewrite the function: Set .
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Swap and : To find the inverse, switch and .
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Solve for : Isolate in terms of .
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First, cube both sides to eliminate the cube root:
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Simplify the equation:
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Solve for :
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Thus, the inverse function is:
Final Answer:
Would you like a more detailed breakdown of the steps or have any further questions? Here are some related topics to explore:
- How to check if two functions are inverses of each other.
- The geometric interpretation of inverse functions.
- How to find the inverse of a function algebraically.
- The concept of domain and range in inverse functions.
- How to find the inverse of non-one-to-one functions.
Tip: Always remember to check if the original function is one-to-one before attempting to find its inverse!
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Cube Roots
Algebra
Formulas
f^-1(x) = 10 - x^3
Theorems
Inverse function theorem
Suitable Grade Level
Grades 9-11