The inverse of a function literally undoes the action of the function if f(2)=3, then f inverse(3)=2. This "f inverse" statement is notated as . Every function has an inverse, but not every inverse will be a function. The inverse may fail the vertical line test. If the original function is a onetoone function, then its inverse will be a function. A onetoone function not only passes the vertical line test, it also passes a horizontal line test.
Onetoone Function  A function or relation is onetoone if no two points in the function have different xcoordinates and the same ycoordinates. (This function then passes a horizontal line test. 
Horizontal Line Test  A test for whether a relation is onetoone. If the relation never has a horizontal line intersect the graph in more than one point, it passes the test and is onetoone. 
Inverse of a Function  The inverse of a function literally undoes the action of a function. Every function has an inverse, but not every inverse will be a function. 
Find the inverse of the function:

Given Function 

Restate the function using y in place of f(x) 

Swap the x and y 

Solve the equation for y. 

The result is the inverse of function f. 
Find the inverse of the function:

Given Function 

Restate the function using y in place of f(x) 

Simplify the right side. 

Swap the x and y 

Solve the equation for y. 

The result is the inverse of function g. 
Find the inverse of the function:

Given Function 

Restate the function using y in place of f(x) 

Swap the x and y 

Solve the equation for y. 

The result is the inverse of function f. 
We now graph each of the functions with their inverse:
Example 1 from above:
We plot the original function in Red, the Identity Function in Green, and the Inverse Function in Yellow. Note that the Inverse Function is symmetric to the original function with respect to the Identity Function. 

Example 2 from above:
We plot the original function in Red, the Identity Function in Green, and the Inverse Function in Yellow and Blue. (positive root in yellow and negative root in blue) Note that the Inverse Function is symmetric to the original function with respect to the Identity Function. 

Example 3 from above:
We plot the original function in Red, the Identity Function in Green, and the Inverse Function in Yellow. Note that the Inverse Function is symmetric to the original function with respect to the Identity Function. 
From Example 3 above, we see that a logarithm is just the inverse of an exponential
function. Whereas all exponential functions went through the point (0,1), we
know that all logarithmic functions will go through the point (1,0). Just as
the base of the exponential function affected the steepness of the exponential
graph, the logarithmic base will also affect the steepness of the logarithmic
graph.
Some Notes from Class
