1. One-to-One Functions;
Section 6.2
One-to-One Functions;
Inverse Functions 1

2. 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.

3. Relations
Doing so produces the graph of R.

4. 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.

5. 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.

6. Inverse Relation - Example
The graph of R and of R-1 (in red).

7. 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

8. 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) }

9. 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.

10. R-1 When R is a Function
f  {(x , y) | y  x2 } f -1 {(x , y) | x  y2 }

11. One to One Function 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.

12. Examples

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

16. Theorem
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.

18. 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}

19. 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}