# 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)} The plots of the set of ordered pairs of function f and its inverse g are shown below. ## Notations and Properties of inverse functionsThe 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 ^{ -1}f 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 ^{-1}_{o} f)(0) = f(f^{ -1}(0)) = f(- 3) = 0
(f ^{-1}_{o} f)(1) = f(f^{ -1}(1)) = f(- 1) = 1
If we continue with the remaining inputs of f, we note that (f ^{ -1}_{o} 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.
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 ^{ -1}_{o} 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. 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. More on one-to-one functions.
## ExercisesExercise 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 ExercisesExercise 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}.
Functions f and h do not have inverses.
Exercise 2:## More References and links on inverse functionsProperties of Inverse FunctionsFind the Inverse Function - Questions Find the Inverse Function (1). Find the Inverse Function (2) Inverse Function - Interactive Tutorial one-to-one functions. |