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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
Find Inverse Of Cube Root Functions.
Find Inverse Of Square Root Functions.
Find Inverse Of Logarithmic Functions.
Find Inverse Of Exponential Functions.
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