Examples, with detailed solutions, on how to find the inverse of exponential functions and also their domain and range.

## Example 1Find the inverse function, its domain and range, of the function given by^{x-3}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^{x-3}
- Take the ln of both sides to obtain
x-3 = 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 2Find the inverse, its domain and range, of the function given by^{(2 x + 3)} + 4
- 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 3Find the inverse, its domain and range, of the function given by^{(x 2 - 1)} + 2 , for x ≥ 0
- 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, + ∞)
## ExercisesFind the inverse, its domain and range, of the functions given below1. 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
More links and references related to the inverse functions. ## More References and Links to Inverse FunctionsFind 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. |