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 f1(x)f^{-1}(x), we need to reverse the process that the function f(x)=3x+73f(x) = \sqrt[3]{3 - x + 7} applies to its input. The general steps are as follows:

Steps to Find the Inverse of f(x)f(x):

  1. Rewrite the function: Set y=f(x)y = f(x). y=3x+73y = \sqrt[3]{3 - x + 7}

  2. Swap xx and yy: To find the inverse, switch xx and yy. x=3y+73x = \sqrt[3]{3 - y + 7}

  3. Solve for yy: Isolate yy in terms of xx.

    • First, cube both sides to eliminate the cube root: x3=3y+7x^3 = 3 - y + 7

    • Simplify the equation: x3=10yx^3 = 10 - y

    • Solve for yy: y=10x3y = 10 - x^3

Thus, the inverse function is: f1(x)=10x3f^{-1}(x) = 10 - x^3

Final Answer:

f1(x)=10x3f^{-1}(x) = 10 - x^3


Would you like a more detailed breakdown of the steps or have any further questions? Here are some related topics to explore:

  1. How to check if two functions are inverses of each other.
  2. The geometric interpretation of inverse functions.
  3. How to find the inverse of a function algebraically.
  4. The concept of domain and range in inverse functions.
  5. 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