Inverses

I.  Quiz 2

II.  Homework Questions

III.  Inverses

Definition:  The inverse of a relation R is the relation consisting of all ordered pairs (y,x) such that (x,y) belongs to R

Example:  The inverse of the relation

(2,3), (4,5), (2,6), (4,6) is

(3,2),  (5,4), (6,2), (6,4)

Generally we switch the roles of x and y to find the inverse.

For functions, we follow the steps below to find the inverse:

Step 1)

Switch the x and y.

Step 2)  Solve for y

Step 3)  Write in inverse notation

Example

Find the inverse of y  = 2x + 1

Solution

1) We write:  x = 2y + 1

2)  We solve:  x - 1 = 2y, y = (x- 1)/2

3)  We write f^-1(x) = (x - 1)/2

Exercises:   Find the inverse of

f(x) = (x - 1)/(x + 1)

f(x) = 3x³ - 2   

f(x) = 3x - 4

III.  Graphing Inverses

To graph an inverse we imaging folding the paper across the y = x line and copy where the ink smeared in the other side.  This will be demonstrated in class.

IV.  One to One Functions

A function  y = f(x) is called one to one if for every y value there is only one x value with

y = f(x).

Example:

y = 2x - 3

For any two values a and b if a = 2x + 3 and b = 3x + 3, then a = b.  

The Horizontal Line Test

If the graph of a function is such that every horizontal line passes through the graph at at most one point then the function is 1-1.  (Examples will be given in class)

If a function is 1-1, then its inverse is also a function.