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 f(x)=1x3f(x) = \sqrt[3]{1 - x}, and we need to find its inverse, f1(x)f^{-1}(x).

To find the inverse of a function, we follow these steps:

Steps to find f1(x)f^{-1}(x):

  1. Rewrite the function with y=f(x)y = f(x):

    y=1x3y = \sqrt[3]{1 - x}

  2. Swap xx and yy:

    x=1y3x = \sqrt[3]{1 - y}

  3. Solve for yy:

    • Cube both sides to eliminate the cube root:

    x3=1yx^3 = 1 - y

    • Solve for yy:

    y=1x3y = 1 - x^3

Thus, the inverse function is:

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

Final Answer:

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

Would you like further clarification or additional details on any of the steps?


Here are some related questions to explore further:

  1. What is the domain and range of f(x)f(x) and its inverse?
  2. How would you verify that f(x)f(x) and f1(x)f^{-1}(x) are indeed inverses?
  3. How does the graph of f(x)f(x) relate to the graph of f1(x)f^{-1}(x)?
  4. What are the general conditions for a function to have an inverse?
  5. 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