Inverse Function Definition

The inverse function definition is explored using graphs. The conditions under which a function has an inverse are also explored.

Explore the Definition of the Inverse

Let function f be defined as a set of ordered pairs as follows:
f = { (-3 , 0) , (-1 , 1) , (0 , 2) , (1 , 4) , (5 , 3)}
The inverse of function f is defined by interchanging the components (a , b) of the ordered pairs defining function f into ordered pairs of the form (b , a). Let g be the inverse of function f; g is then given by
g = {(0 , - 3) , (1 , - 1) , (2 , 0) , (4 , 1) , (3 , 5)}
definition of the inverse of a function using Venn diagrams

Figure 1. The definition of the inverse of a function using Venn diagrams
The plots of the set of ordered pairs of function f and its inverse g are shown below.

definition of the inverse of a function

Figure 2. The definition of the inverse of a function using graphs

Notations and Properties of inverse functions

The inverse of a function f is written as f -1 and its must not be confused with a power.
If function f is defined by: f = { (-3 , 0) , (-1 , 1) , (0 , 2) , (1 , 4) , (5 , 3)}
its inverse f
-1f is defined by interchanging the inputs and outputs as follows.
f
-1 = {(0 , - 3) , (1 , - 1) , (2 , 0) , (4 , 1) , (3 , 5)}
Note that the domain of the inverse f
-1 is the range of f and the range of the inverse f -1 is the domain of f.

What is the inverse function needed for?
In some situations we now the output of a function and we need to find the input and that is where the inverse function is used.
Example: Find x such that 0 < x < π/2 and
sin(x) = 0.2
x = arcsin(0.2) , here arcsin is the inverse of sin(x).

Let us now evaluate the following:
(f
o f -1)(0) = f(f -1(0)) = f(- 3) = 0
(f
o f -1)(1) = f(f -1(1)) = f(- 1) = 1
(f
o f -1)(2) = f(f -1(2)) = f(0) = 2
If we continue with the remaining inputs of f
-1, we note that (f o f -1)(x) = x for all x in the domain of f -1.

We now evaluate the following:
(f
-1o f)(0) = f(f -1(0)) = f(- 3) = 0
(f
-1o f)(1) = f(f -1(1)) = f(- 1) = 1
If we continue with the remaining inputs of f, we note that (f
-1o f)(x) = x for all x in the domain of f.

Below are shown the graphs of f (in blue) and its inverse f
-1 (in red). We note that each each graph is a reflection of the other on the line y = x.
The graphical properties of a function and its inverse.

Figure 3. The inverse of function f is not a function
Conclusion: Some of the most important properties of a function and its inverse are:
1) The domain of f -1 is the range of f
2) The range of f -1 is the domain of f
3) (f -1o f)(x) = x for x in the domain of f
4) (f o f -1)(x) = x for x in the domain of f -1
5) The graphs of f and f -1 are reflection of each other on the line y = x

More on
properties of inverse functions is included in this website.

Which functions have an inverse function (invertible functions) ?

Let function f be defined as a set of ordered pairs as follows:
f = { (-3 , 0) , (-1 , 2) , (0 , 2) , (1 , 4) , (5 , 3)}
The inverse of function f is obtained by interchanging the components (a , b) of the ordered pairs defining function f into ordered pairs of the form (b , a). Let g be the inverse of function f; g is then given by
g = {(0 , - 3) , (2 , - 1) , (2 , 0) , (4 , 1) , (3 , 5)}
Below are shown the Venn diagrams of function f and its inverse g and we note that g is
not a function (input 2 has two outputs -1 and 0). We say that function f is not invertible (does not have an inverse) because it is not a one-to-one functions or only one to one functions have an inverse.
The inverse is not a function using Venn diagram

Figure 4. The inverse of function f is not a function
Below are shown the graphs of f and its inverse g and we note again that g(2) = 0 and g(2) = - 1 and a vertical line would pass by both points (2 , 0) and (2 , -1) and therefore g is not a function.
The inverse is not a function

Figure 5. The inverse of function f is not a function using graphs
More on one-to-one functions.

Exercises

Exercise 1:
a) Find the domain and range of function f defined by
f = {(-4,2),(-3,1),(0,5),(2,6)}
b) Find the inverse function of f and its domain and range.
Exercise 2:
Which of these functions do not have an inverse?
f = {(-1,2),(-3,1),(0,2),(5,6)}
g = {(-3,0),(-1,1),(0,5),(2,6)}
h = {(2,2),(3,1),(6,5),(7,1)}

Answers to Above Exercises

Exercise 1:
a) domain of f = {-4,-3,0,2} and range of f = {2,1,5,6}
b) inverse of f = {(2,-4),(1,-3),(5,0),(6,2)}
domain of inverse of f = {2,1,5,6} and range of inverse of f = {-4,-3,0,2}.
Exercise 2: Functions f and h do not have inverses.

More References and links on inverse functions

Properties of Inverse Functions
Find the Inverse Function - Questions
Find the Inverse Function (1).
Find the Inverse Function (2)
Inverse Function - Interactive Tutorial
one-to-one functions.