Review : Inverse Functions
In the last
example from the previous section we looked at the two functions
 and
 and
saw that
and as noted in that section this means that there is a
nice relationship between these two functions. Let�s see just what that
relationship is. Consider the following evaluations.
In the first case we plugged
 into
 and
got a value of -5. We then turned around and plugged
 into
 and
got a value of -1, the number that we started off with.
In the second case we did something similar. Here we
plugged
 into
 and
got a value of
 ,
we turned around and plugged this into
 and
got a value of 2, which is again the number that we started with.
Note that we really are doing some function composition
here. The first case is really,
and the second case is really,
Note as well that these both agree with the formula for the
compositions that we found in the previous section. We get back out of the
function evaluation the number that we originally plugged into the composition.
So, just what is going on here? In some way we can think
of these two functions as undoing what the other did to a number. In the first
case we plugged
 into
 and
then plugged the result from this function evaluation back into
 and
in some way
 undid
what
 had
done to
 and
gave us back the original x that we started with.
Function pairs that exhibit this behavior are called
inverse functions. Before formally defining inverse functions and the
notation that we�re going to use for them we need to get a definition out of the
way.
A function is called one-to-one if no two values of
x produce the same y. Mathematically this is the same as saying,
So, a function is one-to-one if whenever we plug different
values into the function we get different function values.
Sometimes it is easier to understand this definition if we
see a function that isn�t one-to-one. Let�s take a look at a function that
isn�t one-to-one. The function
 is
not one-to-one because both
 and
 .
In other words there are two different values of x that produce the same
value of y. Note that we can turn
 into
a one-to-one function if we restrict ourselves to
 .
This can sometimes be done with functions.
Showing that a function is one-to-one is often tedious
and/or difficult. For the most part we are going to assume that the functions
that we�re going to be dealing with in this course are either one-to-one or we
have restricted the domain of the function to get it to be a one-to-one
function.
Now, let�s formally define just what inverse functions
are. Given two one-to-one functions
 and
 if
then we say that
 and
 are
inverses of each other. More specifically we will say that
 is
the inverse of
 and
denote it by
Likewise we could also say that
 is
the inverse of
 and
denote it by
The notation that we use really depends upon the problem.
In most cases either is acceptable.
For the two functions that we started off this section with
we could write either of the following two sets of notation.
Now, be careful with the notation for inverses. The �-1�
is NOT an exponent despite the fact that is sure does look like one! When
dealing with inverse functions we�ve got to remember that
This is one of the more common mistakes that students make
when first studying inverse functions.
The process for finding the inverse of a function is a
fairly simple one although there are a couple of steps that can on occasion be
somewhat messy. Here is the process
Finding the Inverse of a Function
That�s the process. Most of the steps are not all that bad
but as mentioned in the process there are a couple of steps that we really need
to be careful with since it is easy to make mistakes in those steps.
In the verification step we technically really do need to
check that both
 and
 are
true. For all the functions that we are going to be looking at in this course
if one is true then the other will also be true. However, there are functions
(they are beyond the scope of this course however) for which it is possible for
only one of these to be true. This is brought up because in all the problems
here we will be just checking one of them. We just need to always remember that
technically we should check both.
Let�s work some examples.
Example
Given
 find
 .
Solution
Now, we already know what the inverse to this function is
as we�ve already done some work with it. However, it would be nice to actually
start with this since we know what we should get. This will work as a nice
verification of the process.
So, let�s get started. We�ll first replace
 with
y.
Next, replace all x�s with y and all y�s
with x.
Now, solve for y.
Finally replace y with
 .
Now, we need to verify the results. We already took care
of this in the previous section, however, we really should follow the process so
we�ll do that here. It doesn�t matter which of the two that we check we just
need to check one of them. This time we�ll check that
 is
true.
|
we
want to find the inverse function,
.
with
y. This is done to make the rest of the process easier.
.
In other words, we�ve managed to find the inverse at this point!
and 
are
both true. This work can sometimes be messy making it easy to make
mistakes so again be careful.
