Inverse Functions
Functions
Inverse
One-to-one
One-to-one
Suppose fff ::: A
A→
B isis aaa function.
function. We
We call
call fff one-to-one
one-to-one ifif every
every distinct
distinct
Suppose
A
→B
B
function.
We
call
pair of
of objects
objects in
in A
A isis assigned
assigned to
to aaa distinct
distinct pair
pair of
of objects
objects in
in B.
B. In
In other
other
pair
A
to
distinct
pair
words, each
target has
has at
at most
most one
one object
object from
from the
the domain
domain
words,
each object
object of
of the
the target
target
has
at
most
one
assigned
assigned to
to it.
it.
previous definition
definition in
in aa more
more mathematical
mathematical
There
There isis aa way
way of
of phrasing
phrasing the
the previous
previous
definition
language:
one-to-one ifif whenever
whenever we
we have
have two
two objects
objects a,a,cc ∈e AA with
with
language:
language: fff isis one-to-one
we
have
~
c,
we are
$ fff(c).
aa �=
(a)
(c).
�= c,
c, we
are guaranteed
guaranteed that
that fff(a)
(a) �=
�=
(c).
—*
Example.
IR →
JR where
= x
not one-to-one
one-to-one because
because 33 �=~ −3
—3
Example.
(x)
Example. fff ::: R
R
→R
R
where fff(x)
(x) =
=
xx222 is
is not
not
and
yet
= f
and fff(—3)
equal 9.
9.
and
(3)
(−3)
(3)
(−3)
and yet
yet fff(3)
(3) =
=
ff(—3)
(−3) since
since fff(3)
(3) and
and
(−3) both
both equal
—*
Horizontal line
line test
test
Horizontal
If aa horizontal
horizontal line
line intersects
intersects the
the graph
graph of
of fff(.x)
in more
more than
than one
one point,
point,
(x) in
If
(x)
is
not
then ff(z)
f(x)
(x) is
is not
not one-to-one.
one-to-one.
then
reason
would not
that the
the graph
graph would
would contain
contain
The reason
reason ff(x)
f(x)
(x) would
not be
be one-to-one
one-to-one is
is that
The
that
have the
the same
for example,
example, (2,
(2,3)
and
two points
points that
that have
same second
second coordinate
coordinate – for
two
3) and
(4,3).
3). That
That would
would mean
mean that
that fff(2)
(2) and
and fff(4)
(4) both
both equal
(4,3).
That
would
equal 3,
3, and
and one-to-one
one-to-one
(4,
(2)
(4)
functions can’t
can’t assign
different
objects in
in the
the domain
can’t
assign two
two different
different objects
domain to
to the
the same
same object
object
functions
of the
the target.
target.
of
every horizontal
horizontal line
in R
R
intersects the
the graph
graph of
If every
line in
JR222 intersects
of aa function
function at
at most
most
If
once, then
then the
the function
function is
is one-to-one.
one-to-one.
once,
once,
—
22
Examples. Below
Below is
is the
of ff ::: R
R
→R
R
where ff(z)
Examples.
(x) =
. There
the graph
graph of
JR →
R where
= x
z2.
There isis aa
horizontal line
line that
that intersects
this graph
graph in
in more
more than
than one
horizontal
intersects this
one point,
point, so
so ff isis not
not
one-to-one.
one-to-one.
—,
\~. )L2
71
66
66
Below is the graph of g R
1W where gQr) = x3. Ally horizontal line that
3
Below
is
the
graph
of
g
:
R
→
R1Wwhere
where
g(x)
=in=x
x33at
Any
horizontal
line
3x3.
Below
isthe
the
graph
of
whereg(x)
gQr)
Ally
horizontal
line
that
could
be is
drawn
would
graph
of
g=
most
one
point,line
so
gthat
is
Below
is
the
graph
ofintersect
RR
→ the
R
where
g(x)
=
xx
Any
horizontal
that
=
Any
horizontal
line
that
Below
graph
of
gg ::gR
→
R
.... Any
horizontal
line
that
could
bebedrawn
drawn
would
intersect
the
graph
ofof
ininat
atatmost
most
one
point,
sosogggg gis
is is
couldbe
drawnwould
wouldintersect
intersectthe
thegraph
graphof
mostone
onepoint,
point,so
one-to-one
could
be
drawn
would
intersect
the
graph
of
in
at
most
one
in
at
most
one
point,
so
is
could
gggg gin
point,
so
is
one-to-one.
one-to-one
one-to-one.
one-to-one.
—~
—~
Onto
Onto
Onto f : A
Onto
Suppose
B is a function. We call f onto if the range of f equals
Suppose
A
→B
BBis
isisaaa afunction.
function.
We
call
onto
the
range
ofoffff fequals
equals
B Suppose
Supposefff f::: A
: A→
function.We
Wecall
callfff fonto
ontoififif ifthe
therange
rangeof
equals
Suppose
A
→
B
is
function.
We
call
onto
the
range
of
equals
B.
In
B other words, f is onto if every object in the target has at least one object
B.
B.
InInother
other
words,
isisonto
onto
if if
every
object
ininthe
the
target
has
atatleast
least
one
object
from
the
domain
assigned
to
it
by f.
other
words,
onto
every
objectin
thetarget
targethas
hasat
leastone
oneobject
object
In
words,
fff fis
ifif
every
object
In
other
words,
is
onto
every
object
in
the
target
has
at
least
one
object
from
the
domain
assigned
to
it
by
f
.
from
fromthe
thedomain
domainassigned
assignedto
from
the
domain
assigned
totoit
ititby
bybyff..f.
Examples. Below is the graph of f 1W
1W where f(x) = z2. Using
2
Examples.
Below
isthe
the
graph
of
R
→
R1Wwhere
where
(x)
=
x22that
Using
2z2.
Examples.
Below
graph
R
... Using
techniques
learned
inis
chapter
“Intro
to1W→
Graphs”,
wefff(x)
can
see
the
Examples.
Below
isthe
the
graphof
offff f::: R
where
f(x)=
=x
Using
Examples.
Below
is
the
graph
of
R
→
R
where
(x)
=
x
Using
techniques
learned
in
the
chapter
“Intro
totoGraphs”,
Graphs”,
wewe
can
see
that
the
techniques
chapter
to
we
range
of f islearned
[0,
oo). in
The
target
of f“Intro
is“Intro
1W, and
[0,Graphs”,
oo) $ 1W
socan
fcan
issee
not
onto.
techniques
learned
inthe
the
chapter
seethat
thatthe
the
techniques
learned
in
the
chapter
“Intro
to
Graphs”,
we
can
see
that
the
range
of
f
is
[0,
∞).
The
target
of
f
is
R,
and
[0,
∞)
=
�
R
so
f
is
not
onto.
rangeof
oo).The
Thetarget
targetof
and[0,
oo)�=
notonto.
onto.
range
range
ofofff fis
isis[0,
[0,[0,∞).
∞).
The
target
ofofff fis
isisR,
R,1W,and
and
[0,[0,∞)
∞)
�=$R
R1Wso
sosoff fis
isisnot
not
onto.
—~
—~
—>
—>
Below
is the
graph
of gof g1W R 1W where
g(x)
= =
x3.x3.The
function
g hasline
thethat
Below
is the
graph
1W where
gQr)
Ally
horizontal
3
3
3
Below
isisthe
the
graph
ofofgequals
R1W→
→
R1Wthe
where
g(x)
=g=xx
xgin3x3.
The
function
has
the
is
of
R
where
g(x)
...isat
The
function
ggg ghas
the
set Below
1WBelow
for its
range.
This
the
target
of
g,of=
so
onto.
thegraph
graph
g::: R
where
g(x)
The
function
has
the
Below
is
the
graph
of
ggintersect
R
→
R
where
g(x)
=
The
function
has
the
could
be
drawn
would
graph
most
one point,
so
g is
set
R
for
its
range.
This
equals
the
target
of
g,
so
g
is
onto.
set
R
for
its
range.
This
equals
the
target
of
g,
so
g
is
onto.
set
forits
itsrange.
range.This
Thisequals
equalsthe
thetarget
targetofofg,g,sosog gisisonto.
onto.
set
R1Wfor
one-to-one
—~
—~
—~
*
Onto *
*
*
*
*** **** **** ****
Suppose
f : A B is
—~
B
*
*
*
*
*
*
*
*
*
*** **** 67**** **** **** *** * *** * *** * *** *
67 We call f onto if the range of
a*function.
72
67
67
67
f
equals
In other words, f is onto if every object in the target has at least one object
from the domain assigned to it by f.
What an inverse function is
Suppose f : A → B is a function. A function g : B → A is called the
inverse function of f if f ◦ g = id and g ◦ f = id.
If g is the inverse function of f , then we often rename g as f −1 .
Examples.
• Let f : R → R be the function defined by f (x) = x + 3, and let
g : R → R be the function defined by g(x) = x − 3. Then
f ◦ g(x) = f (g(x)) = f (x − 3) = (x − 3) + 3 = x
Because f ◦ g(x) = x and id(x) = x, these are the same function. In symbols,
f ◦ g = id.
Similarly
g ◦ f (x) = g(f (x)) = g(x + 3) = (x + 3) − 3 = x
so g ◦ f = id. Therefore, g is the inverse function of f , so we can rename g
as f −1 , which means that f −1 (x) = x − 3.
• Let f : R → R be the function defined by f (x) = 2x + 2, and let
g : R → R be the function defined by g(x) = 12 x − 1. Then
f ◦ g(x) = f (g(x)) = f
�1
2
�
x−1 =2
�1
2
�
x−1 +2=x
Similarly
�
1�
g ◦ f (x) = g(f (x)) = g(2x + 2) =
2x + 2 − 1 = x
2
Therefore, g is the inverse function of f , which means that f −1 (x) = 12 x − 1.
73
The Inverse of an inverse is the original
If f −1 is the inverse of f , then f −1 ◦ f = id and f ◦ f −1 = id. We can see
from the definition of inverse functions above, that f is the inverse of f −1 .
That is (f −1 )−1 = f .
Inverse functions “reverse the assignment”
The definition of an inverse function is given above, but the essence of an
inverse function is that it reverses the assignment dictated by the original
function. If f assigns a to b, then f −1 will assign b to a. Here’s why:
If f (a) = b, then we can apply f −1 to both sides of the equation to obtain
the new equation f −1 (f (a)) = f −1 (b). The left side of the previous equation
involves function composition, f −1 (f (a)) = f −1 ◦ f (a), and f −1 ◦ f = id, so
we are left with f −1 (b) = id(a) = a.
The above paragraph can be summarized as “If f (a) = b, then f −1 (b) = a.”
Examples.
• If f (3) = 4, then 3 = f −1 (4).
• If f (−2) = 16, then −2 = f −1 (16).
• If f (x + 7) = −1, then x + 7 = f −1 (−1).
• If f −1 (0) = −4, then 0 = f (−4).
• If f −1 (x2 − 3x + 5) = 3, then x2 − 3x + 5 = f (3).
In the 5 examples above, we “erased” a function from the left side of the
equation by applying its inverse function to the right side of the equation.
When a function has an inverse
A function has an inverse exactly when it is both one-to-one and onto.
This will be explained in more detail during lecture.
*
*
*
*
*
*
*
74
*
*
*
*
*
*
Using inverse functions
Inverse functions are useful in that they allow you to “undo” a function.
Below are some rather abstract (though important) examples. As the semester continues, we’ll see some more concrete examples.
Examples.
• Suppose there is an object in the domain of a function f , and that
this object is named a. Suppose that you know f (a) = 15.
If f has an inverse function, f −1 , and you happen to know that f −1 (15) = 3,
then you can solve for a as follows: f (a) = 15 implies that a = f −1 (15). Thus,
a = 3.
• If b is an object of the domain of g, g has an inverse, g(b) = 6, and
g (6) = −2, then
b = g −1 (6) = −2
−1
• Suppose f (x + 3) = 2. If f has an inverse, and f −1 (2) = 7, then
x + 3 = f −1 (2) = 7
so
x=7−3=4
*
*
*
*
*
*
*
*
*
*
*
*
*
The Graph of an inverse
If f is an invertible function (that means if f has an inverse function), and
if you know what the graph of f looks like, then you can draw the graph of
f −1 .
If (a, b) is a point in the graph of f (x), then f (a) = b. Hence, f −1 (b) = a.
That means f −1 assigns b to a, so (b, a) is a point in the graph of f −1 (x).
Geometrically, if you switch all the first and second coordinates of points
in R2 , the result is to flip R2 over the “x = y line”.
75
New
New
function
New
function
function
How points in graph of f(x)
How
points
in
of of
new
graph
Howbecome
points points
in graph
graph
of ff(x)
(x)
become
points
of
new
graph
become points of new graph
visual effect
visual
visual effect
effect
f’(x)
ff−1
−1(x)
(x)
(a, b)
(b, a)
(a,
b)
→
�
(b,
a)
(a, b) �→ (b, a)
flip over the “x = y line”
flip
flip over
over the
the “x
“x =
= yy line”
line”
Example.
Example.
Example.
(3,
s),
c(z)
(s, 3’)
(a,-a)
(-3~.-5)
**
**
**
**
**
**
**
71
76
**
**
**
**
**
**
How to find an inverse
If you know that f is an invertible function, and you have an equation for
f (x), then you can find the equation for f −1 in three steps.
Step 1 is to replace f (x) with the letter y.
Step 2 is to use algebra to solve for x.
Step 3 is to replace x with f −1 (y).
After using these three steps, you’ll have an equation for the function
−1
f (y).
Examples.
• Find the inverse of f (x) = x + 5.
Step 1.
y =x+5
Step 2.
x=y−5
Step 3.
f −1 (y) = y − 5
• Find the inverse of g(x) =
2x
x−1 .
Step 1.
y=
2x
x−1
Step 2.
x=
y
y−2
Step 3.
g −1 (y) =
y
y−2
Make sure that you are comfortable with the algebra required to carry out
step 2 in the above problem. You will be expected to perform similar algebra
on future exams.
You should also be able to check that g ◦ g −1 = id and that g −1 ◦ g = id.
77
Exercises
In #1-6, g is an invertible function.
1.) If g(2) = 3, what is g −1 (3)?
2.) If g(7) = −2, what is g −1 (−2)?
3.) If g(−10) = 5, what is g −1 (5)?
4.) If g −1 (6) = 8, what is g(8)?
5.) If g −1 (0) = 9, what is g(9)?
6.) If g −1 (4) = 13, what is g(13)?
For #7-12, solve for x. Use that f is an invertible function and that
f −1 (1) = −2
f −1 (2) = 3
f −1 (3) = 2
f −1 (4) = 5
f −1 (5) = −7
f −1 (6) = 8
f −1 (7) = −3
f −1 (8) = 1
f −1 (9) = 4
7.) f (x + 2) = 5
8.) f (3x − 4) = 3
9.) f (−5x) = 1
10.) f (−2 − x) = 2
11.) f ( x1 ) = 8
5
12.) f ( x−1
)=3
78
Each
the
functions
given
#13-18
invertible.
Find
the
equations
for
Each
Eachofof
ofthe
thefunctions
functionsgiven
giveninin
in#13-18
#13-18isis
isinvertible.
invertible. Find
Findthe
theequations
equationsfor
for
their
inverse
functions.
their
inverse
functions.
Each of
the functions given in #13-18 is invertible. Find the equations for
their inverse
functions.
their inverse functions.
13.)
(x)
3x
13.)
13.)fff(x)
(x)==
=3x
3x++
+222
13.) −x
f(z)+=5 3x + 2
14.)
g(x)
14.)
14.)g(x)
g(x)==
=−x
−x++55
14.)=1g(z)
= —x + 5
1
15.)
h(x)
1
15.)
h(x)
=
x
15.) h(x) =x x
15.) h(z)
xxx
2J
16.)
(x)
16.)
x−1
16.)fff(x)
(x)==
=x−1
x−1
—
2x+3
2x+3
17.)
g(x)
17.)
17.)g(x)
g(x)==
= 2x+3
xxx
2x+3
17.)= g(x)
xxx =
x
18.)
h(x)
18.)
4−x
18.)h(x)
h(x)==4−x
4−x
18.)h(x)=~
19.)
Below
the
graph
→
(0,
∞).
Does
have
an
inverse?
19.)
19.)Below
Belowisis
isthe
thegraph
graphofof
offff: ::RR
R→
→(0,
(0,∞).
∞).Does
Doesfffhave
havean
aninverse?
inverse?
19.) Below is the graph of
f
R
—÷
[0, oo). Does
f
have an inverse?
20.)
Below
the
graph
(0,
∞)
→
R.
Does
have
an
inverse?
20.)
20.)Below
Belowisis
isthe
thegraph
graphofof
ofggg: ::(0,
(0,∞)
∞)→
→R.
R.Does
Doesggghave
havean
aninverse?
inverse?
20.) Below is the graph of g : [0, oo) —÷ 1W. Does g have an inverse?
7974
74
74
−1
−1(x). What are the coordinates
21.)
Below
are
of
and
(x)
and fand
fand
(x).
What
are the
21.) 21.)
Below
are the
the
graphs
of ff(x)
21.)
Below
aregraphs
the graphs
graphs
of f(x)
f(x)
f’(x).
What
are coordinates
the coordinates
coordinates
Below
are
the
of
f’(x).
What
are
the
−1
−1
of
the
points
A
and
B
on
the
graph
of
f
(x)?
of f of
of the
A and
onBBthe
ofpoints
the points
points
and
ongraph
the graph
graph
of(x)?
f’(z)?
of
the
AA B
and
on
the
f’(z)?
‘3‘3
‘xl
‘xl
(“I)
(“I)
AA
,‘
,‘
&2,4)
&2,4)
−1
−1(x).
22.)
Below
are
of
and
What
are
of
g(x)
and gand
gand
(x).
What
are the
the
coordinates
of of
22.) 22.)
Below
are the
the
graphs
of g(x)
Below
are
the
of
g’(x).
What
are
the
22.)
Below
aregraphs
the graphs
graphs
of g(x)
g(x)
g’(x).
What
arecoordinates
the coordinates
coordinates
of
the
C
and
D
on
the
graph
of
g(x)?
the points
points
C
and
D
on
the
graph
of
g(x)?
the
the points
points CC and
and DD on
on the
the graph
graph of
of gQr)?
gQr)?
DD
~x)
~x)
5(X)
5(X)
/
/
/X=t
/X=t
75
80
75
75
75