Relations and Functions OBJ Find the inverse of a relation Draw the graph of a function and its inverse Determine whether the ID: 322205
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Slide1
Inverse Relations and Functions
OBJ:
Find the
inverse
of a
relation
Draw the
graph
of a
function
and its
inverse
Determine whether the
inverse
of a
function
is a
function
Slide2
Original relation
Inverse relation
y
4
2
0
–
2
–
4
x
2
1
0
–
1
–
2
RANGE
F
INDING
I
NVERSES OF
L
INEAR
F
UNCTIONS
x
4
2
0
–
2
–
4
y
2
1
0
–
1
–
2
An
inverse relation
maps the output values back to their
original input values. This means that the domain of the
inverse relation is the range of the original relation and
that the range of the inverse relation is the domain of the
original relation.
RANGE
DOMAIN
DOMAINSlide3
F
INDING
I
NVERSES OF
L
INEAR
FUNCTIONS
x
y
4
2
0
– 2
– 4
2
1
0
– 1
– 2
Original relation
x
4
2
0
– 2
– 4
y
2
1
0
– 1
– 2
Inverse relation
Graph of original
relation
Reflection in
y
=
x
Graph of inverse
relation
y
=
x
– 2
4
4
– 2
– 1
2
– 1
2
0
0
0
0
1
– 2
– 2
1
2
– 4
– 4
2Slide4
F
INDING
I
NVERSES OF
L
INEAR
FUNCTIONS
To find the inverse of a relation that is given by an
equation in x and y, switch the roles of x and
y and
solve for
y (if possible).Slide5
Finding an Inverse Relation
Find an equation for the inverse of the relation
y
= 2
x
– 4
.
y
= 2
x
– 4
Write original relation.
S
OLUTION
Divide each side by
2
.
2
x
+ 2 =
y
1
2
The inverse relation is
y
=
x
+ 2.
1
2
If both the original relation and the inverse relation happen to be
functions, the two functions are called
inverse functions
.
Switch
x
and
y
.
x
y
Add
4
to each side.
4
x
+
4
= 2
y
x
= 2 y
– 4Slide6
Finding an Inverse Relation
I
N
V
E
R
S
E F
U
N C
T
I O
N
S
Functions
f and
g are inverses of each other provided:f
(g (
x)) = x
and g
( f
(x)) = x
The function g is denoted by
f –
1, read as
“f inverse.”
Given any function, you can always find its inverse relation
by switching x and
y. For a linear function f (
x ) = mx
+ b where m
0, the inverse is itself a linear function.Slide7
Verifying Inverse Functions
S
OLUTION
Show that
f
(
g
(
x
)) =
x
and
g
(f
(x)) = x.
g
(
f
(
x)) = g
(2x – 4)
= (2
x – 4) + 2
=
x – 2 + 2
=
x
12
Verify that
f
(x) = 2 x – 4 and
g (x) = x + 2 are inverses.
1
2
f
(
g (x
)) = f x
+ 2
12
(
)
=
2 x
+ 2 – 4 = x + 4 –
4
=
x
1
2
(
)
Slide8
Take square roots of each side.
F
INDING
I
NVERSES OF
N
ONLINEAR
F
UNCTIONS
Finding an Inverse Power Function
Find the inverse of the function
f
(
x
) =
x
2.
S
OLUTION
f
(
x
)
= x
2
y =
x 2
x
= y
2
±
x =
y
Write original function.
Replace original
f
(
x
)
with y.
Switch
x and y.
x
0Slide9
Notice that the inverse of
g
(x) = x
3
is a function, but that the inverse of f (
x) = x 2 is not a function.
If the domain of
f
(
x) = x
2 is restricted
, say to only nonnegative numbers, then the inverse of f
is a function.
F
INDING
I
NVERSES OF
N
ONLINEAR
F
UNCTIONS
The graphs of the power functions
f
(
x
)
=
x 2
and g (x
) = x 3 are
shown along with their reflections in the line y = x.
g
(
x
) = x
3
g (x
) = x
3
f
(x)
=
x
2f (x
) =
x
2
inverse of
g (x
) = x 3
g
–
1(x
)
= x
3
inverse of
f
(
x) = x
2
x =
y 2
On the other hand, the graph of
g
(
x
)
=
x
3
cannot be intersected twice with a horizontal line and
its inverse is a function.
Notice that the graph of
f
(
x
)
=
x
2
can be intersected twice with a horizontal line and that its inverse
is
not
a function.Slide10
F
INDING
I
NVERSES OF
N
ONLINEAR FUNCTIONS
H
O
R
I Z
O
N T
A
L L I
N
E T E
S T
If no horizontal line intersects the graph
of a function
f more than once, then the
inverse of f
is itself a function.Slide11
Modeling with an Inverse Function
A
STRONOMY
Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula.
The volume
V
(in cubic kilometers) of this nebula can be modeled by
V
= (9.01
X 10
26 ) t
3 where
t is the age (in years) of the nebula.
Write the inverse function that gives the age of the nebula
as a function of its volume. Slide12
Modeling with an Inverse Function
S
OLUTION
Write original function.
Isolate power.
Take cube root
of each side.
V
= (9.01
X
10
26
)
t
3
V
9.01
X
10
26
=
t
3
Simplify.
(1.04
X
10
– 9
)
V
= t
3
Volume V can be modeled by V = (9.01 X
10 26 ) t
3
Write the inverse function that gives the age of the nebula
as a function of its volume.
V
9.01
X
10
26
3
=
tSlide13
Determine the approximate age of the Ring Nebula given
that its volume is about 1.5
X
10
38 cubic kilometers.
To find the age of the nebula, substitute 1.5
X
10
38
for
V.
Write inverse function.
Substitute for
V
.
5500
Use calculator.
The Ring Nebula is about 5500 years old.
Modeling with an Inverse Function
S
OLUTION
= (1.04
X
10
–
9
)
1.5 X 1038
3
t
= (1.04
X
10
– 9 )
V3