As with other functions we can use a superscript -1 next to the function name and before the input to indicate the inverse function. Thus the inverse function of
is denoted 
and the inverse function of 
could be denoted 
.
Caution is required because the notation is not always consistent, 
should probably mean 
and likewise 
should mean 
.
However, it is common to see 
used to mean 
.
To avoid these issues we can use the prefix "arc-" to denote inverse.
is the inverse function of 
, the same as 
.
Likewise, 
is the inverse function of 
, the same as 
, and 
is the inverse of 
, the same as 
.
No. Only 1 to 1 functions have inverses. The trig functions are periodic, they are not 1 to 1. To talk about an inverse we need to restrict the domain of the functions so that they are 1 to 1, and hen we can take the inverse of the resulting function. A simple example of this is
. 
is not 1 to 1, because 
. However, if we restrict the domain to 
then the function is 1 to 1, and 
.
Graphically, 
is the parabola opening up. When we restrict the domain we throw out the left arm of the parabola. Then we can find the graph of the inverse by reflecting about the line 
.
We can get
to be 1 to 1 by restricting to 
. In the figure below this portion of the 
curve is shown in red.
The restricted function
has domain 
and range 
. It is 1 to 1.
In the graph below we see the graph of the restricted 
function in red and its reflection through the diagonal is the graph of 
, in green.
The domain and range of a function swap for its inverse, so the domain of 
is 
and the range is 
.
Here is the graph of 
by itself:
Certain values of 
we should know:
.
There are also some unexpected results of restricting the domain of 
to get an inverse. It is not always true that 
. For example,
To get
to be 1 to 1 we restrict the domain to 
.
The restricted function
has domain 
and range 
. It is 1 to 1.
In the figure below we reflect the restricted 
curve, in red, to get the graph of 
, in green
So we get the graph of 
:
The domain of 
is 
and the range is 
.
We can make
1 to 1 by restricting to 
.
The restricted function
has domain 
and range 
. It is 1 to 1.
Reflect:
and we get the graph of 
:
has domain 
and range 
.
Date: 2008/10/01 19:33:03