Inverses

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

x - 1
y  =
2

3.  We write

x - 1
f -1(x)   =
2

Notice that the original function took x, multiplied by 2 and added 1, while the inverse function took x, subtracted 1 and divided by 2.  The inverse function does the reverse of the original function in reverse order.

Exercises

Find the inverse of

1.               x - 1
f(x)  =
x + 1

2. f(x)  =  3x3 - 2

3. f(x)  =  3x - 4

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.

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).  That is, each y value comes from a unique x value.

Example

1. Determine if

y = 2x - 3

is one to one

Solution

For any two values a and b if

y = 2a - 3

and

y = 2b - 3

then

2a - 3  =  2b - 3

2a  =  2b

a  =  b

Hence this function is 1-1.

2. Determine if

y  =  x2

is one to one

Solution

Now if

a2   =   b2

then

a   =   b

In particular

a = -b

is a viable solution, for example

12  = (-1)2

hence this function is not 1-1.

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.

The graph below is the graph of a 1 -1 function since every horizontal line crosses the graph at most once.

However, the graph below is not the graph of a 1 -1 function, since there is a horizontal line that crosses the graph more than once.

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

An application of this is if we want a computer to find an inverse function, then we first have the computer check to see if the function is one to one, then have it proceed to find the inverse.

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