Inverse Functions

 

          Inverse functions are functions that “undo” the operations of others, just as addition undoes subtraction. A notable property of inverse function is the following:

 

F(G(x)) = G(F(x)) = x

 

In fact, if this can be shown about two functions, those functions are said to be inverses of each other.

 

Example  Let  f(x) = 2x – 1  while g(x) = x + 1.

                                                               2

 

f (g(x)) = 2(x +1) – 1                                              g(f(x)) =  (2x – 1) + 1

2                                                                                                                                                                                            2

 

           = x + 1 – 1                                                            = 2x

                                                                                            2

           =  x                         

                                                                                        = x

 

 

In both cases where each function is composed into the other function, the result is x. Therefore, the functions, f and g, are inverses of each other. This process verifies that the functions are inverses. In class we said this can be likened to turning on a light: first you plug in the lamp, and then you must flip the switch to get light. The same is true here: it requires both compositions resulting in x to say that the functions are inverses of each other.

 

          Now graph the two functions in your calculator. You should see a cool relationship between these functions: the intersection point lies on the line y = x , which tells you that the graphs of inverse functions form a mirror image over this line. This information should lead you to another conclusion about inverse functions: the domain and range of each one will be switched with the other.

 

Suppose a function  h(x) = {(2. 4). (-6. -12),  (4.4, 0.56). (7, 29)}.

 

We write the inverse as h-1(x) = {(4, 2), (-12, -6), (0.56, 4.4), (29, 7)}.

 

In conclusion, the properties of inverse functions are:

1.     the graphs form a mirror image over the line y = x ;

2.     the domain and range of the functions is switched;

3.     f(g(x)) = g(f(x)) = x

 

 

Now that the properties of inverse functions have been outlined, how do we find the inverse of a function? The first thing to ask is: Does the function have an inverse?

 

To make this determination, we must decide whether the function is 1 – 1. What does it mean to be a 1 – 1 function? The definition states the following:

 

a relation is a 1 – 1 function if every domain is paired with exactly one range, AND, every range is paired with exactly one domain.

 

It is somewhat easy to determine this characteristic by examining the graph of the function and applying what is called the horizontal line test (HLT). Hopefully, you are familiar with the vertical line test (VLT) for functions: dropping a vertical line anywhere along the graph of a relation and touching only one point tells us that graph represents a function. The HLT works the same way, except we draw a horizontal line looking for it to touch only one point on the graph at a time. The graph of a function that can pass the HLT, then, is 1 – 1. This characteristic tells us that the function has an inverse or that the inverse for the particular function exists.

 

So, if a function passes the HLT, how do we find that function that is its inverse?

 

F(x) = 5x – 10 is a 1 – 1 function and, thus, we know it has an inverse. Here’s what you do:

 

Replace F(x) with y, the standard symbol for the range. Switch the domain and range (switch x for y and y for x), and solve for y. The resulting function is the inverse of F.

 

F(x) = 5x – 10

 

y = 5x – 10

 

x = 5y – 10

 

x – 10 = 5y

 

x – 10 = y = F-1(x)

   5

 

Note: This notation F-1 has nothing to do with the reciprocal of F.

 

One further note about 1 – 1 functions. In class we looked at several examples of functions that are NOT 1 – 1: the quadratic y = x2 is a good example. If we look at only part of the quadratic from x > 0, then the function IS 1 – 1, or if we look at the quadratic from x < 0, the same is true. Recall that this process is restricting the domain.