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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