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)(x^{3} + 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 x^{2} + 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 x^{2} + 8 is given by [8,+∞)
which gives a range of ^{3}√(4 x^{2} + 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 x^{2} + 8) + 2

Which may be written as
y  2 = ^{3}√(4 x^{2} + 8)

Cube both side and simplify
(y  2)^{3} = ( ^{3}√(4 x^{2} + 8) )^{3}
(y  2)^{3} = 4 x^{2} + 8

Solve for x
x^{2} = (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}√(x^{2} + 2x + 4) ; x ≥  1 (Hint: start by finding the vertex of x^{2} + 2x + 4 to understand the given domain x ≥  1)
Answers to above exercises
1. f^{ 1}(x) = x^{3} + 3 ; domain: (∞ , ∞) Range: ( ∞ , ∞)
2. g^{ 1}(x) = 1+√(x^{3}  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.
