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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

.
14.18
Definition (Bijective.)
Let

be sets. A function

is called
bijective if and only if

is both injective and 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