1.5: Inverse Funtions

The Inverse of a Function

So in a normal function, the setup would be



However, with an inverse function, the x and y variables are swapped. So now you have...




So as you can see, the inverse of a function is found by switching x with y, and then isolating the y variable.

Another important note, is that the domain of the original function, is the same as the range of the inverse of said function, and the range of the original function, then becomes the domain of the inverse.

This can be shown simply with a graph.


The domain of the original function is All real numbers, and the range is all real numbers greater than zero, however when the inverse of the function is graphed, you find that the range is all real numbers, and the domain is all real numbers greater than zero.


When an inverse equation is graphed, it is compared to the original function by flipping it over the line     

A good example to show this is the graph of    



The inverse is flipped over the line        which shows that they are inverse of one another.

A good way to find out if two functions are inverses is to verify that both

     and        both   

For example

   and     

First plug them into each other

          and                  


Then Solve. In both functions, the 3's add up to 0 and the 2's divide out leaving youu with x in both.

          and         



One-to-One Functions

Definition:  A function      is considered one-to-one if, for a and b in its domain,

   implies   


There are two methods to figure out if a function is one-to-one. You can solve the problem algebraically or graphically.

Algebraic








You don't need to give "b" a  because thats redundant.


Graphic





To know if a function is one-to-one, you must know that there can only be one x output for every y input, and also only one y output for every x output.

This graph is a one-to-one function because if you use the horizontal line test, you can see that the function will only cross the line once.



In conclusion, the inverse of a function is found by first replacing f(x) with y, and then interchanging x and y.


Example

              Write original equation

                    Replace   with y

             

                  Solve for y

        Replace y by