Math Problem Statement

Find a formula for the inverse of the following function, if possible.

g(x)=x^1/3−1

Solution

The function given is:

g(x)=x1/31g(x) = x^{1/3} - 1

To find the inverse function g1(x)g^{-1}(x), follow these steps:

  1. Replace g(x)g(x) with yy: y=x1/31y = x^{1/3} - 1

  2. Solve for xx in terms of yy: y+1=x1/3y + 1 = x^{1/3}

    Now cube both sides to eliminate the cube root: (y+1)3=x(y + 1)^3 = x

  3. Express xx in terms of yy: x=(y+1)3x = (y + 1)^3

  4. Replace yy with xx to get the inverse function**: g1(x)=(x+1)3g^{-1}(x) = (x + 1)^3

Thus, the inverse of the function is:

g1(x)=(x+1)3g^{-1}(x) = (x + 1)^3

Do you have any questions about this solution, or would you like more details?

Here are five related questions:

  1. How can you verify that two functions are inverses of each other?
  2. What is the domain and range of the function g(x)=x1/31g(x) = x^{1/3} - 1?
  3. How does the process of finding the inverse differ for non-linear functions?
  4. What happens if a function doesn't have an inverse? How do you check?
  5. What are some practical applications of inverse functions?

Tip: To check if your inverse is correct, compose the original function with the inverse g(g1(x))g(g^{-1}(x)) and g1(g(x))g^{-1}(g(x)). Both should give you xx.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Inverse Functions
Cube Roots

Formulas

Inverse function formula g^{-1}(x)
Cube root property

Theorems

Inverse Function Theorem
Properties of Cube Roots

Suitable Grade Level

Grades 10-12