Find the Inverse of Cube Root Functions

This tutorial explains how to find the inverse of cube root functions, as well as how to determine their domain and range. Each example is solved step by step using clear algebraic reasoning. Throughout the tutorial, cube roots are written using standard mathematical notation.

Example 1

Find the inverse function, its domain, and its range for

Solution

• Domain and range of the given function:
  \[D = (-\infty, +\infty), \quad R = (-\infty, +\infty)\]
• Write the function as an equation:
  \[y = \sqrt[3]{2x - 1}\]
• Cube both sides:
  \[y^3 = 2x - 1\]
• Solve for x:
  \[x = \tfrac{1}{2}(y^3 + 1)\]
• Interchange x and y:
  \[f^{-1}(x) = \tfrac{1}{2}(x^3 + 1)\]

The domain and range of the inverse function are the same as those of the original function: \[(-\infty, +\infty)\]

Example 2

Find the inverse function, its domain, and its range for

Solution

• Domain and range:
  \[D = (-\infty, +\infty), \quad R = (-\infty, +\infty)\]
• Write as an equation:
  \[y = \sqrt[3]{\tfrac{x}{3} - 1} - 4\]
• Rearrange:
  \[\sqrt[3]{\tfrac{x}{3} - 1} = y + 4\]
• Cube both sides:
  \[\tfrac{x}{3} - 1 = (y + 4)^3\]
• Solve for x:
  \[x = 3\big((y + 4)^3 + 1\big) = 3(y + 4)^3 + 3\]
• Interchange variables:
  \[f^{-1}(x) = 3(x + 4)^3 + 3\]

The inverse function also has domain and range \(( -\infty, +\infty )\).

Example 3

Find the inverse function, its domain, and its range for

Solution

• Because the domain is restricted to \(x \ge 0\), the function is one-to-one.
• Domain and range of the given function:
  \[D = [0, +\infty), \quad R = [4, +\infty)\]
• Write the function as an equation:
  \[y = \sqrt[3]{4x^2 + 8} + 2\]
• Rearrange and cube both sides:
  \[(y - 2)^3 = 4x^2 + 8\]
• Solve for x:
  \[x^2 = \tfrac{1}{4}\big((y - 2)^3 - 8\big)\]
• Select the positive root (since \(x \ge 0\)):
  \[x = \tfrac{1}{4}\sqrt{(y - 2)^3 - 8}\]
• Interchange variables:
  \[f^{-1}(x) = \tfrac{1}{4}\sqrt{(x - 2)^3 - 8}\]

The domain and range of the inverse are: \[D_{f^{-1}} = [4, +\infty), \quad R_{f^{-1}} = [0, +\infty)\]

Exercises

Find the inverse function, its domain, and its range for:

1. \(f(x) = -\sqrt[3]{-x + 3}\)
2. \(g(x) = \sqrt[3]{x^2 + 2x + 4}, \quad x \ge -1\)

Answers

1. \(f^{-1}(x) = x^3 + 3, \quad D = (-\infty, +\infty), \; R = (-\infty, +\infty)\)
2. \(g^{-1}(x) = -1 + \sqrt{x^3 - 3}, \quad D = [\sqrt[3]{3}, +\infty), \; R = [-1, +\infty) \)

Additional resources on inverse functions:

• Find the Inverse of a Cubic Function – Step by Step
• Inverse Function Calculator
• Applications of Inverse Functions
• Inverse of Square Root Functions
• Inverse Logarithmic Functions
• Inverse Exponential Functions