Section 10.6 Inverse functions
We now have enough machinery to start working on inverse functions. Consider what an inverse function needs to do. If we start with and apply a function then we obtain some element We want our inverse to “undo” this and take back to We’ll start by defining functions that are nearly inverses and give a few examples.
But before we do that, we note that some of the material below is a bit technical. The reader who is interested in these details should continue on. The reader who wishes to just get to the main results should instead skip ahead to Definition 10.6.6 and Theorem 10.6.8
So notice that if we start at some point and apply to get then a left-inverse tells us how to get back from to
On the other hand, if are trying to get to a particular point in the codomain using then the right-inverse tells you a possible starting point
Example 10.6.2.
Then notice that satisfies
By swapping the roles of in the above we can construct a function that is a right-inverse but not a left-inverse. Consider the functions defined in the image below.
Notice in the above example, that the function that has a left-inverse is injective, while the function with the right-inverse is surjective. This is not a coincidence as the following two lemmas prove.
Lemma 10.6.3.
Proof.
We prove each implication in turn.
- Assume that
has a left-inverse, Now let so that Then but since is the left-inverse of we know that and Thus and so is injective. - Now let
be injective. We construct a left-inverse of in two steps. First pick any Then consider the preimage of a given point That preimage, is empty or not.- If
then define - Now assume that
Since is injective, the preimage contains exactly 1 element. To see why, consider both in the preimage. We must have (since they are both in the preimage of ), but since is an injection, we must have So define the unique element in the preimage.
To summariseNow let under it maps to some with Hence (as argued above) is the unique element in the preimage of and so Thus is a left-inverse of
Lemma 10.6.4.
Proof.
We prove each implication in turn.
- Assume that
has a right-inverse, Now let and set Then and so is surjective. -
Now let
be surjective and let For the sake of this proof, let us denote the preimage of asNow let then under it maps to some so that (by construction) Hence and thus is a right-inverse as required.
These lemmas tell us that a function has both a left inverse and right inverse if and only if it is bijective. We can go further those one-sided inverses are actually the same function.
Lemma 10.6.5.
Let have a left-inverse, and a right inverse then Further, the function has a left and right inverse if and only if is bijective.
Proof.
Let be as stated. Then we know that
Now starting with we can write:
and thus as required.
The last part of the lemma follows by combining the previous two lemmas.
So this lemma tells us conditions under which a function will have both a left- and right-inverse, and that those one-sided inverses are actually the same function. A function that is a left- and right-inverse is a (usual) inverse.
Definition 10.6.6.
Note that we will prove that the inverse is unique, and so we will be able to say that is the inverse of and denote it
Also note that if a function is an inverse then it is also a left- and right-inverse.
Lemma 10.6.7.
If a function has an inverse, then that inverse is unique.
Proof.
This proof is very similar to the proof of Lemma 10.6.5. Let and both be inverses to the function Then
and so Thus the inverse is unique.
We can now state our main theorem about inverse functions.
Theorem 10.6.8.
Proof.
We combine some of the lemmas above to prove this result.
-
Assume that
has an inverse. Then that inverse is both a left-inverse and a right-inverse. Lemma 10.6.5 then implies that is both injective and surjective, and so is bijective.Now assume that is bijective. Then Lemma 10.6.5 tells us there exists a function that is a left-inverse and right-inverse for Then, by definition is an inverse for - The uniqueness of the inverse is proven by Lemma 10.6.7.
Theorem 10.6.8 tells us under what circumstances a function has an inverse. However, it does not tell us if that inverse has a nice expression. If the original function is nice enough, then we may be able to state the inverse nicely. Here are a couple of such examples.
Example 10.6.9.
The function defined by is bijective and so has an inverse function. We proved this in Result 10.4.2 and Result 10.4.2. The inverse is and we can work out a formula for it by solving for in terms of Notice that we did exactly that when we proved that was surjective. In particular, we found that
Example 10.6.10. Möbius continued.
That one can do this is not as obvious as it might seem. In particular, if is infinite we need the Axiom of Choice in order to make this selection. The interested reader should search engine their way to more information on this. Now in the event that our function is injective, then contains exactly one element and we don’t need the Axiom of Choice to construct our function. Thankfully we apply this result when our function is bijective — phew.