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 = ± (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
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