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 these are very special functions. Let’s see just what makes them so
special. 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. This is a fairly simple definition of
one-to-one but it takes an example of a function that isn’t one-to-one to show
just what it means. Before doing that
however we should note that this definition of one-to-one is not really the
mathematically correct definition of one-to-one. It is identical to the mathematically correct
definition it just doesn’t use all the notation from the formal definition.
Now, let’s see an example of 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 a tedious and
often difficult. For the most part we
are going to assume that the functions that we’re going to be dealing with in
this section are one-to-one. We did need
to talk about one-to-one functions however since only one-to-one functions can
be inverse functions.
Now, let’s formally define just what inverse functions
are.
Inverse Functions
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 is 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.
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 section
if one is true then the other will also be true. However, there are functions (they are far
beyond the scope of this course however) for which it is possible for only 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.
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Example 1 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.
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Example 2 Given
find ,
.
Solution
Now the fact that we’re now using instead of doesn’t change how the process works. Here are the first few steps.
Now, to solve for y
we will need to first square both sides and then proceed as normal.
This inverse is then,
Finally let’s verify and this time we’ll use the other one
just so we can say that we’ve gotten both down somewhere in an example.
So, we did the work correctly and we do indeed have the
inverse.
Before we move on we should also acknowledge the
restrictions of
that we gave in the problem statement but
never apparently did anything with.
Note that this restriction is required to make sure that the inverse, given above is in fact one-to-one.
Without this restriction the inverse would not be
one-to-one as is easily seen by a couple of quick evaluations.
Therefore, the
restriction is required in order to make sure the inverse is one-to-one.
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The next example can be a little messy so be careful with
the work here.
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Example 3 Given
find .
Solution
The first couple of steps are pretty much the same as the
previous examples so here they are,
Now, be careful with the solution step. With this kind of problem it is very easy
to make a mistake here.
So, if we’ve done all of our work correctly the inverse
should be,
Finally we’ll need to do the verification. This is also a fairly messy process and it
doesn’t really matter which one we work with.
Okay, this is a mess.
Let’s simplify things up a little bit by multiplying the numerator and
denominator by .
Wow. That was a lot
of work, but it all worked out in the end.
We did all of our work correctly and we do in fact have the inverse.
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There is one final topic that we need to address quickly
before we leave this section. There is
an interesting relationship between the graph of a function and its inverse.
Here is the graph of the function and inverse from the first
two examples. We’ll not deal with the
final example since that is a function that we haven’t really talked about
graphing yet.
In both cases we can see that the graph of the inverse is a
reflection of the actual function about the line . This will always be the case with the graphs
of a function and its inverse.