### Lecture 29, Composite Functions, One-to-One, Inverse - Math-UMN

```MATH
1051
Lecture
April 7,2017
29
Announcements
5. I &
5.2
Today
Read Section
5.3
for
next
time
5.1
Composite
Functions
Functions
Some
Define
f (× )
×2+
=
) flymeans
functions
¥+1
,
g
-
Review
e.g.
(
(4) 2+24-+1
x
)
3fF2
=
( replace every
but don't
We
can
do
math
to
get
a
value
f ( 4)
"
x
"
with
"
change anything
=
16
+
4
"
else )
's +1=17.5
goes
lgtftfniinminanh.tmpowariii.in
Functions
of
Instead
plug
In
f (× )
×
We
g(
xth
We
)
=
did
number
a
like
,
2+2×+1
)=tH't
ftx
in
expression
an
=
plugging
IT
did
g
this
to
-
(
x
( replace
+1
to
3(x+h)-2f
this
Review
-
test
find
the
4
X
=
even
every
We
,
Can
xth
or
zxfz
"
every
for
( replace
)
like
,
/
"
difference
x
"
odd
x
"
'
with
'
×
-
"
)
.
with
"
Xth
quotient
"
)
Composition of
f(×)=
So
,
×
what
f(g( A)
2+2×+1
woud
=
(
fcgc xD
)2t
g
mean
#
?
+
1
Functions
(
x
)
=
zxfz
Composition of
So
,
f(×)=
×2+
what
woud
fcgc A)
Say
Called
Can
"
f
=
3€
¥+1
fcgc xD
g
(
x
)
=
?
3*22+(3×2.7)
(
of
+
g
of
the
write
Mean
Functions
×
"
ition
fcgc xD
1
OR
#
of
fog
meaning
Jame
fandg
( ×)
fog (
Find
fk
FI
)=2X
fog
Q
(x)
find
=
(
×
)
=
,
(×)
f( gcx) )
=
2
( ×+÷ )
goh ( x )
A) 2×3.1×27
B) 21¥13
(
EI
Find
hogk )
hlgcx ) )=2
Q
Find
)
X2
g
fog
find
×÷
x
#
hof
'
hk )=
=
2×3
.
,2€
# )2C)ak+D3#xn2 D) ?
Hx÷i)±24¥ ) x¥p=¥T5x¥y
-
( ×)
A) 2×3-2×2 B) 4×3-2×2
C) 16×3-4×2
D) ?
fog
Find
fog
is
a
function
too
,
,
so
we
# )
can
find
fog
fog C- 2)
f
( ×)
fog (8)
=
f(g(
gof
732
Q Find
A)
3×2
=
g
8) )
(
If
D=
=3 ( FF 2=3
)
( 2)
2=3.4--12
(3)
B) 33/32
C)
3
D) ?
(2)
.
or
fog (4)
or
f 09 #)
Find
find
¥
goft⇒
jkeo.lt
•
( 0,4
)
f
Looking for
(× )
g(ft5D
'
"ta¥e±⇒
Then
g(
x
)
find
g(o)=4
goftst
glo )
.
4
.
of Composite
Domain
h (x)
Vx
=
g ( ht D
Find
EI
glhk ) )
-1
not
is
(
EI
g (x)
,
the
in
because
Find
-1
g(
g ( hl 4) )
4
is
b( ecause
EI
2
(
II
event
in
hl 4)
=2
find
g(
in
the
though
h
not
is
=
not
(a) )
,
in
#
the
Of
=
NOT
-
¥
2 is not
in
of
in
~~
glh
the
-
Uh oh !
of h (xD
to
=
of
¥
=
domain
domain
¥2
1)
g(h(xD !
the
=
and
domain
2 is
glhl
.
domain
( 4) )
h
¥2
=
#
=
function
uh
oh !
glh ( xD !
the
1.7
of
domain
g(xD
hooray !
( x ))
domain
of g ( x )
!
Wacky! )
of Composite
Domain
E±
f
(
x
)
Find
( domain
¥32
=
h)
=
of
domain
the
of
h (x)
Foh
+1>-0
2-1
×
×
( domain
of
f)
:
hlx )
-31=0
hk)±3
Fit
X
¥3
+
X
I
1
t
9
8
{111×2-1,1*8}
At
(
x
)
.
function
f
Find
EI
Find
A) fk
B)
f
functions
fk
)
=
;
(
KI
)=I
x
)
&
g
gfx )
;
g(x)=
and g
such
(x )
xt
=
KI
that
fog
(× )
=
¥1
f (g ( x ) )
When
EI
f
( x)
X
gof
x
3f2X )
)
=
(x
)
So
,
fog
Show that
fog (
g ( ×)
2X
=3
=
-2
fog (
x
)
( x)
=
-
=
goof
=
6X
(3/1)=-6/1
=
g of
(×)
-
.
( x)
.
=
gcfcx))
5.2
One
-
to
-
One
,
Inverse
Functions
One
functions
ALL
.
:
to
One
-
output
One
Functions
per input /
Not
Function
y=rx
if
X=4
,y=2
One
Fro
-
to
-
other
if
functions
one
:
y=20R
,
-2
(
oneinputperoutputlltgirnizofetsaf
Not
EYE
's
/
'¥i¥
et
#
×=4
test
function
a
y=±rx
-
One-to-One
line
Vertical
two
one
inputs
to
-
.
one
-
one
output
Functions
Inverse
f ( × ) =×3
/
Graphs
Fx
an
f.
g(×)=F×
Reflection
:
Algebra
Called
×3
between
Relationship
:
#
inverse
fogk )
=
x
=
gofcx)
and
functions
If
the
across
,
=
line
(
X
ftp.X
write
f(x)=×3
-
f-
"
,
y=x
Identity
.
'
( x)
.
( × ) =3f×
function )
Must
have
To
one
EI
-
to
an
one
-
One Beto
function
inverse
(
or
"l
-
l
' '
NOT
1-
1
f
(
x
)
One
must
be
)
f- ( x ) ? ?
'
flxkxzqg
-
.
Refle
Ct
across
g
Not
a
function
y=x
Verify
functions
Inverse
Operations
E±:
fory
Solve
Capture
do the
"
or
Inverses
Function
Function
"
Opposite of
that
that
adds
:
"
undo
FK )
subtracts
3
is
subtracting }
=x+3
3 !
×)
"
solving equations
when
6-3=9
=
adding
3
Can
6
y+3=
"
we
way
"
opposite
y
The
the
g(
=
X -3
.
Verify
EI
fog
Verify
(x)
( x 3)
=
go F ( x )
that
-
=
(g
f- ( × )
×
=
( x+3 ) -3
=
'
gk )
+3
Inverses
=
So
×
g
EI
Let
gk )
=
tx
glhlxll
hlg (A)
and
hlx
)=4x
'T ( 4x )
=
=
4
( tax)
x
=x
inverse
function of f)
'
=
f- ( × )
Verify
,
.
=
(× )
the
is
.
that
g-
'
k)
=hk )
.
Find
To
find
f-
( ×)
'
the
Replace
4
X
3
=3y
4
-
X
-
4
34
Find
.
+4
X=3y
2
the
f- ( x )
'
=y
=
¥
in
y
variables
×
&y
y
Replace y
3×+4
y=
1
for
Solve
3×+4
=
with
Interchange
3
fcx )
fk )
equation
1
,
2
EI
Inverse
'
f- ( × )
with
'
f- ( × )
.
in
the
equation
Find
Q
flx )
@
2
3
4
=
¥4
the
,
Inverse
Find
f-
'
(x)
.
Find
E±
The
of
graph
f-
'
( ×)
of
the
flx )
Inverse
is
shown
.
Draw
.
y=fK )
the
graph
Restrict
about
What
inverse
an
If
find
of
1
2
3
4
to
XZO
×=y2
the
fx=y
'
(x)
Nol
1- I
doesn't
so
,
have
.
gk )
y=x2
g-
?
gcx )=×2
function
EI
f(x)=×2
restrict
we
Domain
the
now
,
function
inverse
1-
a
I
/
XZO
,
have
we
xzo
,
yzo
,
(
=V×
no
±
( fk )
because
70
it
but
restricted
we
we
because
don't
rx
need
is
to
always
to
write
201
yzd
```