# Find Inverse Of Cube Root Functions

Find the inverse of cube root functions as well as their domain and range; examples with detailed solutions. In what follows, the symbol 3√ is used to indicate the principal cube root.

 Example 1: Find the inverse function, its domain and range, of the function given by f(x) = 3√(2 x - 1) Solution to example 1: The domain D and range R of the given function are given by: D: (- ∞ , + ∞) and R: (- ∞ , + ∞) In order to find the inverse, we first write the function as an equation as follows y = 3√(2 x - 1) Then solve it starting by cubing both sides y 3 = ( 3√(2 x - 1) ) 3 Simplify and solve for x y 3 = 2 x - 1 x = (1 / 2)(y 3 + 1) Change x into y and y into x to obtain the inverse function. f -1(x) = y = (1 / 2)(x3 + 1) The domain and range of the inverse function are respectively the range and domain of the given function f. Hence domain and range of f -1 are given by: domain: (- ∞ , + ∞) range: (- ∞ , + ∞) Example 2: Find the inverse, its domain and range, of the function given by f(x) = 3√(x / 3 - 1) - 4 Solution to example 2: The domain and range of the given function are given by D: (- ∞ , + ∞) and R: (- ∞ , + ∞) Write the given function as an equation. y = 3√(x / 3 - 1) - 4 which can be writeen as: 3√(x / 3 - 1) = 4 + y Cube both sides of the above equation and simplify. ( 3√(x / 3 - 1) ) 3 = (4 + y) 3 (x / 3 - 1) = (4 + y) 3 Solve for x. x / 3 = (4 + y) 3 + 1 Which gives x = 3 ( (4 + y) 3 + 1 ) = 3 (4 + y) 3 + 3 Interchange x and y to obtain the inverse function f -1(x) = y = 3 (4 + x) 3 + 3 The domain and range of the inverse function are respectively the range and domain of the given function f. Hence domain and range of f -1 are given by: domain: (- ∞ , + ∞) range: (- ∞ , + ∞) Example 3: Find the inverse, its domain and range, of the function given by f(x) = 3√(4 x2 + 8) + 2 ; x ≥ 0 Solution to example 3: Although the formula of the given function indicates an even function (not a one to one), the explicit given domian (x ≥ 0) make the given function a one to one. Domain of f: [ 0 ; ∞) , given. Range: For x in the domain [ 0 ; ∞) , the range of 4 x2 + 8 is given by [8,+∞) which gives a range of 3√(4 x2 + 8) in the interval [3√8 , +∞) or [2 , +∞) and finally, taking into account the shift +2, the range for the given function is given by [4 , +∞) To find the inverse, we first write the given function f as an equation y = 3√(4 x2 + 8) + 2 Which may be written as y - 2 = 3√(4 x2 + 8) Cube both side and simplify (y - 2)3 = ( 3√(4 x2 + 8) )3 (y - 2)3 = 4 x2 + 8 Solve for x x2 = (1 / 4) ( (y - 2)3 - 8 ) x = ~+mn~ (1 / 4) √ ( (y - 2)3 - 8 ) The domain of f is given by [ 0 ; ∞) and therefore x is selected to be given by x = (1 / 4) √ ( (y - 2)3 - 8 ) Interchange x and y to obtain the inverse function f -1(x) = y = (1 / 4) √ ( (x - 2)3 - 8 ) The domain and range of f -1 are respectively given by the range and domain of f found above Domain of f -1 is given by: [4 , + ∞) and its range is given by: [0 , + ∞) Exercises: Find the inverse, its domain and range, of the functions given below 1. f(x) = - 3√(- x + 3) 2. g(x) = 3√(x2 + 2x + 4) ; x ≥ - 1 (Hint: start by finding the vertex of x2 + 2x + 4 to understand the given domain x ≥ - 1) Answers to above exercises: 1. f -1(x) = x3 + 3 ; domain: (-∞ , ∞) Range: (- ∞ , ∞) 2. g -1(x) = -1+√(x3 - 3) ; domain: [3√3 , +∞) Range: [-1 , +∞) More links and references related to the inverse functions. Find the Inverse of a Cubic Function - Step by Step Worksheet. Find the Inverse of a Cubic Function. Find the Inverse Functions - Calculator Applications and Use of the Inverse Functions Find the Inverse Function - Questions Find the Inverse Function (1) - Tutorial. Definition of the Inverse Function - Interactive Tutorial Find Inverse Of Cube Root Functions. Find Inverse Of Square Root Functions. Find Inverse Of Logarithmic Functions. Find Inverse Of Exponential Functions.