Next: 14.4 The Exponential Function
Up: 14. The Inverse Function
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14.12
Definition (Injective.)
Let
![$A$](img6.gif)
and
![$B$](img145.gif)
be sets. A function
![$f: A \to B$](img3498.gif)
is called
injective
or
one-to-one if and only if for all points
![$a,b$](img152.gif)
in
or equivalently if and only if
If
![$f$](img676.gif)
is a function whose domain and codomain are subsets of
R then
![$f$](img676.gif)
is injective if and only if each horizontal line
intersects the graph of
![$f$](img676.gif)
at most once.
14.13
Examples.
Let
![$f:[0,\infty) \to \mbox{{\bf R}}$](img3502.gif)
and
![$g:\mbox{{\bf R}}\to \mbox{{\bf R}}$](img3503.gif)
be defined by
Then
![$f$](img676.gif)
is injective, since for all
![$x,y \in [0,\infty)$](img3506.gif)
we have
![$x+y>0$](img3507.gif)
,
and hence
However
![$g$](img722.gif)
is not injective, since
![$g(-1) = g(1).$](img3509.gif)
14.14
Remark (Strictly monotonic functions are injective.)
If
![$h$](img50.gif)
is strictly increasing on an interval
![$J$](img1131.gif)
, then
![$h$](img50.gif)
is injective on
![$J$](img1131.gif)
,
since for all
Similarly, any strictly decreasing function on
![$J$](img1131.gif)
is injective.
14.15
Definition (Surjective.)
Let
![$A,B$](img659.gif)
be sets and let
![$f: A \to B$](img3498.gif)
. We say that
![$f$](img676.gif)
is
surjective if and only if
![$B = \mbox{image}(f)$](img3515.gif)
, i.e. if and only if for every
![$b \in B$](img3516.gif)
there
is at least one element
![$a$](img31.gif)
of
![$A$](img6.gif)
such that
![$f(a) = b$](img3517.gif)
.
14.16
Examples.
Let
![$f:\mbox{{\bf R}}\to \mbox{{\bf R}}$](img3518.gif)
and
![$g : \mbox{{\bf R}}\to [0,\infty)$](img3519.gif)
be defined by
Then
![$g$](img722.gif)
is surjective, since if
![$x \in [0,\infty),$](img3522.gif)
then
![$x = g(\sqrt x)$](img3523.gif)
, but
![$f$](img676.gif)
is not surjective, since
![$-1$](img1465.gif)
is not
in the image of
![$f$](img676.gif)
.
14.18
Definition (Bijective.)
Let
![$A,B$](img659.gif)
be sets. A function
![$f: A \to B$](img3498.gif)
is called
bijective if and only if
![$f$](img676.gif)
is both injective and surjective.
14.20
Remark.
Let
![$A$](img6.gif)
and
![$B$](img145.gif)
be sets, and let
![$f: A \to B$](img3498.gif)
be a bijective function.
Let
![$b$](img32.gif)
be a generic element of
![$B$](img145.gif)
. Since
![$f$](img676.gif)
is surjective, there is an element
![$a$](img31.gif)
in
![$A$](img6.gif)
such that
![$f(a) = b$](img3517.gif)
. Since
![$f$](img676.gif)
is injective this element
![$a$](img31.gif)
is unique, i.e. if
![$a$](img31.gif)
and
![$c$](img57.gif)
are elements of
![$A$](img6.gif)
then
Hence we can define a function
![$g:B \to A$](img3529.gif)
by the rule
Then by definition
Now let
![$a \in A$](img725.gif)
, so that
![$f(a) \in B$](img3532.gif)
. It is clear that the
unique
element
![$s$](img66.gif)
in
![$A$](img6.gif)
such that
![$f(s) = f(a)$](img3533.gif)
is
![$s=a$](img3534.gif)
, and hence
14.21
Definition (Inverse function.)
Let
![$A,B$](img659.gif)
be sets, and let
![$f: A \to B$](img3498.gif)
. An
inverse function for
![$f$](img676.gif)
is a function
![$g:B \to A$](img3529.gif)
such that
14.22
Remark (Bijective functions have inverses.)
Notice that in the definition of inverse functions, both the domain
and the codomain of
![$f$](img676.gif)
enter in a crucial way.
It is clear that if
![$g$](img722.gif)
is an inverse function for
![$f$](img676.gif)
, then
![$f$](img676.gif)
is an
inverse function for
![$g$](img722.gif)
.
Remark
14.20
shows that every bijective function
![$f: A \to B$](img3498.gif)
has an inverse.
14.23
Example.
Let
![$f:[0,\infty)$](img3537.gif)
be defined by
We saw above that
![$f$](img676.gif)
is bijective, and hence has an inverse. If
Then it is clear that
![$g$](img722.gif)
is an inverse function for
![$f$](img676.gif)
.
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
![$E$](img165.gif)
denote the inverse of the logarithm
function. Thus
![$E$](img165.gif)
is a function from
R to
![$\mbox{${\mbox{{\bf R}}}^{+}$}$](img711.gif)
, and it
satisfies the conditions
We will investigate the properties of
![$E$](img165.gif)
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
![\begin{displaymath}
\Big(f(s) = f(t) \Big) \mbox{$\hspace{1ex}\Longrightarrow\hs...
... \mbox{$\hspace{1ex}\Longrightarrow\hspace{1ex}$}\Big(s=t\Big)
\end{displaymath}](img3564.gif) |
(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