Section 6.2

One-to-One Functions;

Inverse Functions

Relations

Definition: A relation is a set of points, (ordered

pairs) in the plane.

As an example, consider the relation

R {(2 , 1), (4 , 3), (0 , 3 )}

As written, R is described using the roster method.

Since R consists of points in the plane, we follow

our instinct and plot the points.

Relations

Doing so produces the graph of R.

Inverse Relation

Definition: If R is a relation, then the relation

R-1 {(y , x) | (x , y) R }

is called the inverse relation of R.

The inverse relation R-1 is obtained from R by

interchanging the x and y for every point in R.

Inverse Relation - Example

As an example, consider the relation

R {(2 , 1), (4 , 3), (0 , 3 )}

The inverse relation R-1 is given by

R-1 {(1, 2 ), (3 , 4), (3 , 0 )}

Notice that the graph of R-1 is obtained from the

graph of R by reflecting about the line y x all the

points in R.

The graph of R and of R-1 (in red).

Inverse Relation - Example

Domain and Range of R-1

Definition: If R is relation, then the relation

R-1 {(y , x) | (x , y) R }

is called the inverse relation of R.

From the definition follows that,

Dom R-1 Ran R

Ran R-1 Dom R

R-1 When R is a Function

If a relation f is a function defined by an equation

of the form y f (x), that is, if

f {(x , y) | y f (x) }

then, the inverse relation f -1 of f is defined by the

new equation x f (y), that is,

f -1 {(x , y) | x f (y) }

R-1 When R is a Function

In this case, the fact that f is a function does not

automatically imply that f -1 is also a function.

For instance,

f {(x , y) | y x2 }

is a function but

f -1 {(x , y) | x y2 }

is not.

R-1 When R is a Function

f {(x , y) | y x2 } f -1 {(x , y) | x y2

}

The previous discussion leads to the following

definition:

This condition guarantees that the inverse relation

f-1 of a one to one function f is also a function.

In this case, f-1 is called inverse function of f.

One to One Function

Examples

For each function, use the graph to determine whether the function is one-to-one.

A function that is increasing on an interval I is a one-to-one function in I.

A function that is decreasing on an interval I is a one-to-one function on I.

Theorem

To find the inverse we interchange the elements of the domain with the elements of the range.

The domain of the inverse function is {0.8, 5.8, 6.1, 6.2, 8.3}

The range of the inverse function is {Indiana, Washington, South Dakota, North Carolina, Tennessee}

Find the inverse of the following one-to-one function:{(-5,1),(3,3),(0,0), (2,-4), (7, -8)}

State the domain and range of the function and its inverse.

The inverse is found by interchanging the entries in each ordered pair:

{(1,-5),(3,3),(0,0), (-4,2), (-8,7)}

The domain of the function is {-5, 0, 2, 3, 7}

The range of the function is {-8, -4,0 ,1, 3). This is also the domain of the inverse function.

The range of the inverse function is {-5, 0, 2, 3, 7}

1

1

1

3 3Verify that the inverse of is 5.

5

For what values of is ?

For what values of is ?

f x f xx x

x f f x x

x f f x x

Note the domain of is 5 and the domain of is 0 .f x x g x x

1 3

35

5

x

f f x

3 55 provided 5

3

xx x

1 3

35 5

x

f f x

3 provided 0

3x x

x

1 11 12 3 3 2 3 3

2 2f f x x x x f f x