# Core 3 Notes Edexcel

notes for core 3, edexcel exam board but i think they generally overlaps with a few other exam boards

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• Created by: tom
• Created on: 06-06-11 14:07

First 678 words of the document:

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
n g
any
fact
ors
w hi
ch
are
common
to
bot
h t
he nume r
ator
and t
he denominator
.
Al
gebrai
c Fract
ions
can be
simpli
fi
ed by
cancel
li
ng down.
To do t
his,
numerat
ors
and
denomi
n at
ors
mu st
be f
ull
y
factor
ised
fi
rst
.
If
t
here
are
fract
i
o ns
w i
thi
n t
he
numerat
or/
d enominator
,
mu l
ti
ply
by a
commo n f
act
or t
o get
ri
d of
t
hese and
creat
e an
equi
val
ent f
ract
ion:
1 x+1 ( 1 x+1)×6 3(x+2)
2
1 2 = ( 2
1
x+ 2 )×6
= 3x+6 3
2x+4 = 2(x+2) = 2
3 x+ 3 3 3
To
mu l
ti
ply
fr
acti
ons,
simpl
y mul
ti
ply
the
nume r
ator
s and
mu l
ti
ply
the
denomina t
ors.
I
f
possi
ble,
cancel
down
fi
rst
.
To
divi
de by
a f
ract
ion,
mul
ti
ply
by t
he reci
procal
of
t
he
f
ract
ion:
x+1 3 3(x+1)
2 × x2-1 = x+1 3 3
2 × (x+1)(x-1) = 2(x+1)(x-1) = 2(x+1)
To
subt
ract
f
ract
ions,
t
hey must
have
the
same denomi
nat
or.
Thi
s i
s done
by
f
indi
ng
the
lowest
commo n mult
i
p l
e of
t
he
denominat
ors:
1 4 1(x+6) 4(x+1) (x+6)+4(x+1) 5x+10
(x+1) + (x+6) = (x+1)(x+6) + (x+6)(x+1) = (x+1)(x+6) = (x+1)(x+6)
When
the
nume rat
or
has t
he same
or
a hi
gher
degr
ee
than
the
denominator
(
it
i
s
an
i
mproper
f
ract
ion)
,
you c
a n
divi
de
the
ter
ms t
o pr
oduce
a mi
xed f
ract
ion:
Funct
ions
Funct
ions
are speci
al
types
of
mappings
su ch
that
ever
y el
eme nt
of
t
he domain
is
ma pped
to
exact
l
y one el
eme nt
i
n
the
range.
Thi
s i
s i
l
lustr
ated
below f
or
the
funct
ion
f
(x)
=
x +
2

## Other pages in this set

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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
ange
of
a f
uncti

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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 two
di
ff
erent
val
ues i
n t
he r
a nge.
Theref
ore
y = ± x i
s
not
a f
unct

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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
ion
has
produced
the
number
9,
but
we
do
not
know what
i
nput
pro du ce d t
h i
s nu mb er

### 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 l
u s
s ig n, i
.e . a ll v alue s a r
e p o s itiv e .
e.

### 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
ved
graphi
cal

### 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 ca n n ot find a n exact sol
uti
on t
o the e q ua t

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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
f
unct
i
o n
is
negati

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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 ) = x
e
is
a speci
al
number
si
mil
ar
to .
It
has
a
v
a l
ue of

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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)
only
have
an i
nverse
funct
ion
if
t
heir
doma ins are
restr
ict
ed so
tha t
they ar
e
on etoone functi