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# 14.3 Inverse Functions

14.12   Definition (Injective.) Let and be sets. A function is called injective or one-to-one if and only if for all points in

or equivalently if and only if

If is a function whose domain and codomain are subsets of R then is injective if and only if each horizontal line intersects the graph of at most once.

14.13   Examples. Let and be defined by

Then is injective, since for all we have , and hence

However is not injective, since

14.14   Remark (Strictly monotonic functions are injective.) If is strictly increasing on an interval , then is injective on , since for all

Similarly, any strictly decreasing function on is injective.

14.15   Definition (Surjective.) Let be sets and let . We say that is surjective if and only if , i.e. if and only if for every there is at least one element of such that .

14.16   Examples. Let and be defined by

Then is surjective, since if then , but is not surjective, since is not in the image of .

14.17   Exercise. A Give examples of functions with the following properties, or else show that no such functions exist.

, is injective and surjective.

, is injective but not surjective.

, is surjective but not injective.

, is neither injective nor surjective.

14.18   Definition (Bijective.) Let be sets. A function is called bijective if and only if is both injective and surjective.

14.19   Examples. If is defined by

then is bijective.

The function is a bijective function from to R. We know that ln is strictly increasing, and hence is injective. If is any real number we know that takes on values greater than , and values less that , so by the intermediate value property (here we use the fact that is continuous) it also takes on the value , i.e. is surjective.

14.20   Remark. Let and be sets, and let be a bijective function. Let be a generic element of . Since is surjective, there is an element in such that . Since is injective this element is unique, i.e. if and are elements of then

Hence we can define a function by the rule

Then by definition

Now let , so that . It is clear that the unique element in such that is , and hence

14.21   Definition (Inverse function.) Let be sets, and let . An inverse function for is a function such that

14.22   Remark (Bijective functions have inverses.) Notice that in the definition of inverse functions, both the domain and the codomain of enter in a crucial way. It is clear that if is an inverse function for , then is an inverse function for . Remark 14.20 shows that every bijective function has an inverse.

14.23   Example. Let be defined by

We saw above that is bijective, and hence has an inverse. If

Then it is clear that is an inverse function for .

We also saw that is bijective, and so it has an inverse. This inverse is not expressible in terms of any functions we have discussed. We will give it a name.

14.24   Definition (.) Let denote the inverse of the logarithm function. Thus is a function from R to , and it satisfies the conditions

We will investigate the properties of after we have proved a few general properties of inverse functions.

In order to speak of the inverse of a function, as we did in the last definition, we should note that inverses are unique.

14.25   Theorem (Uniqueness of inverses.) Let be sets and let
. If and are inverse functions for , then .

Proof:     If and are inverse functions for then

and

Also for all

(I have used the facts that for all , and for all ).

14.26   Theorem (Reflection theorem.) Let be a function which has an inverse function . Then for all

Proof:     Let be a function that has an inverse function . Then for all

and

Thus

Now

and

and the theorem now follows.

Remark: If is a bijective function with and codomain() Then the reflection theorem says that if is the inverse function for , then graph() = (graph) where is the reflection about the line .

Since we know what the graph of looks like, we can make a reasonable sketch of graph.

It is a standard notation to denote the inverse of a function by . However since this is also a standard notation for the function which is an entirely different object, I will not use this notation.

We have shown that if is bijective, then has an inverse function. The converse is also true.

14.27   Theorem. Let be sets and let . If has an inverse function, then is both injective and surjective.

Proof:     Suppose has an inverse function . Then for all in we have
 (14.28)

and hence is injective. Also, for each

so image(), and is surjective.

Next: 14.4 The Exponential Function Up: 14. The Inverse Function Previous: 14.2 Applications   Index
Ray Mayer 2007-09-07