Notes, Lesson 2.5
Inverse Functions and Logarithms
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Finding the Inverse of a Function | Definition | Graphing a Function and its Inverse | Check Concepts

Finding the Inverse of a Function

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 one-to-one function, then its inverse will be a function. A one-to-one function not only passes the vertical line test, it also passes a horizontal line test.

Definitions


One-to-one Function A function or relation is one-to-one if no two points in the function have different x-coordinates and the same y-coordinates. (This function then passes a horizontal line test.
Horizontal Line Test A test for whether a relation is one-to-one. If the relation never has a horizontal line intersect the graph in more than one point, it passes the test and is one-to-one.
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.

Graphing a Function and its Inverse

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.

Logarithms

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

Check Concepts
Check Concepts
Check Concepts


#1: True or False: The "-1" in the notation:  does not just indicate reciprocal.
   
#2: True or False: One-to-one Functions pass the "horizontal line test."
   
#3: True or False: Exponential functions are the inverse of logarithmic functions.
   
#4: How do inverses appear on a graph in relation to the original function?
   
#5 True or False: If an inverse is a function, then the original must be a function

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