Inverse Functions

Another form of a function is an Inverse function.  An Inverse function is written as

Two functions, f and g are Inverses of each other if 

 and  

This is true because Inverse functions undo each other

An inverse function is formed by interchanging the "x" and "y" variables in the equation. To find an Inverse function, one must follow these steps:

 Step 1- Interchange variables

 Step 2- Isolate the "y" algebraically 

Example:

    Original Function- 

                              

                   Step 1

                   

                   Step 2

                                  

                             

Algebraically Verifying Inverse Functions

To verify Inverse functions algebraically, the original definition of the function is used (see above)

An example of this is shown below, where   and  are proven to be true with each function given.

Example:

          

Note that the solution of each composition function is x, meaning that function g is the Inverse of function f.

Graphically Verifying  Inverse Functions

There is also a graphical way to look at an Inverse function. In comparison to the original function, Inverse functions are flipped over the line y = x. An example of a function (f) and its Inverse (g) are shown on the graph below 

Example:

Determining If an Inverse is a Function

Not every function has an Inverse though (an Inverse that is a function). 

A function's Inverse is not a function if it is not one-to-one.  A function is one-to-one if and only if implies  

An example of a function that is not one-to-one is a parabola. Below is a graph of a parabola and it's Inverse in blue and red, respectively. 

Example:

Note that blue parabola has two y values when x = 1, -1 and 1. This implies that the function is not one to one. 

Algebraically Determining One-to-One

To algebraically find if a function has an Inverse, then it must be determined if the function is one to one by applying the definition. To determine this, the variables a and b must be plugged into the original equation and set equal to each other, as seen below.

Example:

Is the function one-to-one?

Since a = b, the function is one-to-one. 

Graphically Determining One-to-One

If only the Inverse of a function is given, the way to determine if it is an one-to-one function is to evaluate the function with the horizontal line test. The horizontal line test is the exact same as the vertical line test, except for that the function is tested with a horizontal line. This proves the function is one-to-one because for every y there is only one x value if the function passes the test. An example is shown below.

Example:

Domain and Range of Inverse Functions

Another important thing to note is that the domain of an Inverse function is the range of the original function. For more information on domain and range, click here. 

Example:

If  has a data table that looks like this

 

then  has a domain of 2, 5, 12, and 18. 

For More Information on this Topic

http://www.purplemath.com/modules/invrsfcn.htm

https://www.khanacademy.org/math/algebra/algebra-functions/function_inverses/v/introduction-to-function-inverses

Work Cited

Community Wiki. PSTricks Parabola and Inverse. Digital image. Tex. StackExchange, 2012. Web. 7 Jan. 2015. <http://tex.stackexchange.com/questions/72712/plot-inverse-function-in-pgfplots>.

Emathhelp. Horizontal Line Test. Digital image. EmathHelp. EmathHelp, 2010. Web. 7 Jan. 2015. <http://www.emathhelp.net/notes/calculus-1/function-concept/inverse-of-a-function/>.

Sparknotes. Graphing Parabolas. Digital image. SparkNotes. SparkNotes, 2014. Web. 7 Jan. 2015. <http://www.sparknotes.com/math/algebra1/quadratics/section1.rhtml>.