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Page 1

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Al
gebr
aic
Fr
act
i
o ns
An
algebrai
c f
ract
i
o n
can al
w ays
be expr
essed
in
differ
ent
,
yet
equival
ent
for
ms.
A
f
ract
ion
is
expressed
in
it
s si
mp l
est
for
m by
cancell
i…

Page 2

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The
set
of
a l
l
numb ers
that
we can
feed i
nto
a funct
ion
is
call
ed t
he
doma i
n of
the
funct
ion.

The
set
of
a l
l
numb ers
that
t
h e
funct
ion
produces i
s cal
led
the r…

Page 3

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For
exampl
e,
consi
der
t
he
expr
essi
on

y = ±x .




Noti
ce
that
any val
ue
o f
x
in
the
doma i
n,
e xc ept
x = 0
,
(
i.
e.
any posi
ti
ve r
eal
numb er)
i
s mapped to…

Page 4

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Consi
der
t
he
simpl
e,
li
near
f
unct
ionf (
x)
=
3x
27.
I
f
w e
feed
x =
2 i
nto
thi
s f
unct
ion,
we
get

out
f (
2)
=
21.
Suppose
that
we
are
tol
d t
hat
t
he
funct…

Page 5

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The
Modul
us
Funct
ion
The mo du lu s sig n in d i
c ate s tha t
w e ta k e t
h e ab s o lu t e va l
u e o f
th e expr
essi
on
insi
de
the
mod u…

Page 6

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To
sol
ve
the
equat
ion
|ax + b| = |cx + d|
it
i
s easi
est
t
o squar
e bot
h si
des
to
remove
the
modul
us si
gn.
Equat
ions
of
the
for
m |ax + b| = cx + d
need
to
be
sol…

Page 7

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Numer
ical
Met
hods
I
n rea l lif
e si
tuat
ions , w e ar e o ften
faced wi
th
equa ti
o n s
w h i
c h ha ve n o ana l
ytic
s o l
ution.

Th at is to
say we…

Page 8

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t
hat
t
he
sol
uti
on
is
somewher
e ar
ound
x = 0.7
.
We
noti
ce t
h at
t
o t
he
lef
t
of
the
root
,
the
funct
ion
is
posi
ti
ve
and
to
the
ri
ght
of
t
he
root
t
he…

Page 9

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Exponent
ial
and
Log
Funct
ions
The
exponent
ial
f
uncti
on
(e) a n d the
n a t
u ra l
logar
it
hm
funct
ion
(l
n)
are
bot
h t
he
inver
se
oper
ati
ons
of
one anot
her.
eln(
x) = ln x
(e…

Page 10

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Tr
igonomet
ry
For
o ne to one f
uncti
ons,
you
can draw i
ts i
nver
se graph by r
efl
ecti
ng i
t
in
the
li
ne y
= x.

The three
tr
ig
funct
ions
sin(
x ),
cos (
x)
an d t
an(x)…

Comments

Ayman!

Can't read a lot of the formulae on Mac :(

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