or equivalently if and only if

If is a function whose domain and codomain are subsets of

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

However is not injective, since

Similarly, any strictly decreasing function on is injective.

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

, is injective and surjective.

, is injective but not surjective.

, is surjective but not injective.

, is neither injective nor surjective.

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.

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

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.

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.

. If and are inverse functions for , then .

and

Also for all

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

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.

(14.28) |

so image(), and is surjective.