Inverse Functions

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A function  is said to be one-to-one  if

     whenever     

Here is a one-to-one function

While the function below is not one-to-one.

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Horizontal Line test  

A function is one-to-one if and only if no horizontal line intersects its graph more than once. The animation below shows that the function  is one-to-one.

 

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If we reverse the inputs and outputs of the function F  below,

we get a new function G  as follows .

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Notice that for all in the domain of F , we have

 

and for all in the domain of G , we have

.

Let  be a one-to-one function with domain A  and range B . Then the inverse function  has domain B and range A and is defined by

for any in B.

This says that if  maps to , then its inverse function maps back to .

In the example above , G  is the inverse function of F .

Does function H have an inverse function? Why?

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How to find the inverse function of a one-to-one function  ?

Step 1   Write .
Step 2   Solve this equation for in terms of .
Step 3  To express as a function of , interchange and , the resulting equation is .

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For example, if we consider the function . ( Note that is one-to-one.)

Solving the equation for in terms of ,

we get that the inverse function is   .
The following figure shows the graphs of   (red) and (blue). Note that they are mirror images of each other through the line (green).

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Now we consider what does it mean to invert the sine function. First we note that there is a complication with trigonometric function that did not arise with the functions we have just discussed. The sine function clearly fails the horizontal line test.

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The following figure shows two periods of the sine function (blue) and the result (red) of interchanging and in the equation .

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The problem is that the red curve is obviously not the graph of a function. In order to have an invertible function that takes all the values of the sine function, we restrict the domain of the sine to the interval from to .

This restricted sine and its inverse (called arcsine , in Maple we use the command arcsin(x) ) are shown in the next figure.

(red)

(blue)

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Here are problems to think about:

1. If we restrict the domain of sine to [ , ], what does the resulting inverse function look like? What is the relation between this inverse function and ?

By restricting the domain of cosine to the interval from to , we invert cosine in the same fashion. The resulting inverse function is called arccosine. Note that the domain of both and are [ , 1 ].

2. What is the relation between and ?

If and , the exponential function is either increasing or decreasing and so it is one-to-one. It therefore has an inverse function which is called the logarithmic function with base and is denoted by . Hence

and

  for every ,
   for every .

Laws of Logarithms

1.  
2.
3.       (where is any real number)

The logarithm with base is called the natural logarithm  and has special notation

  .

Hence, we have

    for every ,
    for every .

In particular,

  .

The following figure shows the graphs of   (red) and (blue). Note that they are mirror images of each other through the line (green).

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