Example 1
Find the inverse function, its domain and range, of the function given by
f(x) = e^{x3}
Solution to example 1

Note that the given function is a an exponential function with domain (∞ , + ∞) and range (0, +∞).
We first write the function as an equation as follows
y = e^{x3}

Take the ln of both sides to obtain
x3 = ln y or x = ln y + 3

Change x into y and y into x to obtain the inverse function.
f^{ 1}(x) = y = ln x + 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: (0,+ ∞) range: (∞ , + ∞)
Example 2
Find the inverse, its domain and range, of the function given by
f(x) = 2 e^{(2 x + 3)} + 4
Solution to example 2

Let us first find the domain and range of the given function.
Domain of f: (∞ , + ∞)
Range: for x in the domain, the range of e^{(2 x + 3)} is given by (0,+∞)
The range of 2 e^{(2 x + 3)} is also given by (0,+∞)
The range of f(x) = e^{(2 x + 3)} + 4 is (4,+∞) because the +4 shifts up the graph of the function

Find the inverse of f, write f as an equation and solve for x.
y = 2 e^{(2 x + 3)} + 4
2 e^{(2 x + 3)} = y  4
e^{(2 x + 3)} = (y  4)/2
Take the ln of both sides to obtain
2x + 3 = ln ((y  4)/2)
and finally x = (1/2) (ln ((y  4)/2)  3)

Change x into y and y into x to obtain the inverse function.
f^{1}(x) = y = (1/2) (ln ((x  4)/2)  3)
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: (∞ , + ∞)
Example 3
Find the inverse, its domain and range, of the function given by
f(x) = 2 e^{(x 2  1)} + 2 , for x ≥ 0
Solution to example 3

It is easy to show that function f given by the formula above is an even function and therefore not a one to one if the domain is R. However the domain in our case is given by x ≥ 0 which makes the given function a one to one function and therefore has inverse.
Domain of f: [0 , + ∞) , given
Range: for x in the domain [0, + ∞) , the range of x^{ 2} is given by [0,+∞) which can be written as
x^{ 2} ≥ 0
subtract 1 to both sides to obtain: x^{ 2}  1≥  1
take the exponential of both sides to obtain: e^{x 2  1 } ≥ e^{ 1} (the exponential function being an increasing function)
multiply by +2 to both sides of the above inequality to obtain: 2 e^{x 2  1 } ≥ 2 e^{ 1}
add +2 to both sides of the above inequality to obtain: 2 e^{x 2  1 } + 2≥ 2 e^{ 1} + 2
the left hand side of the above inequality is the given function, hence the range of the given function is given by : [2 e^{ 1} + 2, + ∞)

Find the inverse of f, write f as an equation and solve for x.
y = 2 e^{(x 2  1)} + 2
2 e^{(x 2  1)} = y  2
e^{(x 2  1)} = (y  2)/2
Take the ln of both sides to obtain
x^{ 2}  1 = ln ((1/2)(y  2))
and finally x = + or  sqrt[ln ((1/2)(y  2)) + 1]
Since x ≥ 0 (given domain), we have x = sqrt[ln ((1/2)(y  2)) + 1]

Change x into y and y into x to obtain the inverse function.
f^{1}(x) = y = sqrt[ln ((1/2)(x  2)) + 1]
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: [2 e^{ 1} + 2, + ∞) and its range is given by: [0, + ∞)
Exercises
Find the inverse, its domain and range, of the functions given below
1. f(x) = e^{x + 4}
2. g(x) = 2  e^{(4x  2) / 3}
3. h(x) =  e^{(2 x 2  5) } + 3, for x ≤ 0
Answers to above exercises
1. f^{ 1}(x) = ln( x)  4 ; domain: (∞ , 0) Range: (∞ , +∞)
2. g^{ 1}(x) = (3/4) ln (2  y) +1/2 ; domain: (∞ , 2) Range: (∞ , +∞)
3. h^{ 1}(x) =  sqrt[(1/2) ln (3  y) + 5/2]; domain: (∞ ,  e^{(5)} + 3) Range: (∞ , +∞)
More links and references related to the inverse functions.
More References and Links to Inverse Functions
Find the Inverse of a Rational Function  Step by Step Worksheet
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.
