Mathematics 3200
Unit 2 Lesson 5
Inverse of a Relation
Focus on…

Sketching the graph of the inverse of a relation

Determining if a relation and its inverse are functions

Determining the equation of an inverse
Inverse Of A Function

If f is a function with domain A and range B , the inverse function, if it exists, is denoted by f

1 and has domain B and range A
 f

1 maps y to x if and only if f maps x to y
Horizontal Line Test

A test used to determine if the graph of an inverse relation will be a function

If it is possible for a horizontal line to intersect the graph of a relation more than once, then the inverse of the relation is not a function
Example 1: Consider the graph of the relation shown.
A)
B) Sketch the graph of the inverse relation.
Determine whether the relation and its inverse are functions.
C) State the domain, range, and intercepts for the relation and the inverse relation.
D) State any invariant points.
A) The relation is a function of x because there is only one value of y in the range for each value of x in the domain. In other words, the graph of the relation passes the vertical line test.
The inverse relation is not a function of x because it fails the vertical line test.
There is more than on value of y in the range for at least one value of x in the domain. You can confirm this by using the horizontal line test on the graph of the original relation.

B) To graph the inverse relation, interchange the x
 coordinates and y
 coordinates of key points on the graph of the relation.
Point on the Relation


6 , 0



4 , 4
 

0 , 4
 
 
Points on the Inverse Relation

0 ,

6


4 ,

4
 

4 , 0
 
 
The graphs are reflections of each other in the line y
 x . The points on the graph of the relation are related to the points on the graph of the inverse relation by the mapping
   
.
C) The domain of the relation becomes the range of the inverse relation and the range of the relation becomes the domain of the inverse relation.
The x
 intercept of the relation becomes the y
 and the y
 intercept of the inverse relation intercept of the relation becomes the x
 intercept of the inverse relation.
Relation
 x |

6
Domain
 x

6 , x
 

 x | 0
 x

 y
4 , x
 
  y
| 0

Range
|

6
 y y

4 , y
 


6 , y
 

Inverse
Relation
D) The invariant point is
 
. x-intercept


6 , 0

 
y-intercept
 

0 ,

6
