Calculus I ( Michealmas)

, 1. Functions
1. 1 Functions domain and
,
range

i.
Definition function : A function f is a
correspondence between

two sets D I called the domain ) and < ( called the codomainl ,




that each element 17 and element of
assigns to of one
only
um
one


C


Notation : f :D HE


2.
Definition range
: the set of all
images is called the
range of f-
Notation : Rauf =
If :
XE Domf }


Domain &
Range
3. :




• For X E D , f- IX1E C


FIX)
× :
argument of f : the value of f- at X

( or the
image of × under ft

. Notation i.
such that statement
,



f- : DHL :
X t> f- 1×1

f is a
function from 17 to C that associate × in 17 to fix, in C




4.
Examples :




f IR IR HX2 f- takes IR to IR by f- 1×1=5
i
×
given
: :
,




Domf =
IR TX E IR
tail
Rauf =
[a ,
oo
)


2
f- 1×1 =
12×+4 ,
XE [a. 6 ]


Domf =
[ 0,6J


f- lol = 2 , f- 161 =
4

i.
Rauf =
[ 2,4J

,Default : . Take ← domain to be IR

.
Take domain to be the maximum set of real number ×
mum


st f- IX1 is a real number

Questions :



I
f- 1×1=12×+4

Domf = [ -2,0 ]

Rauf = [ a ,
oo ]


2
f- 1×1 = Itx
Damf = IR1 { I } = I -0,11 v It ,
oo )
"
IR takes out 1

Rauf = IR1 to } =
f- a ,o}v So , so }
Common
5. Notation :
YIX)
X -



independent variable

y
-



dependent variable

'
'
E.
g i FIX) =
X ,
y= ×

i.
f- 1×1 =y
'

So ,
the function f- IX1 is
defined by the
equation y=X

'
2
f- IX1 =
X ,
y =
×


When Xao has No real solutions
y
•

,



-
when × so ,
y
=
FX .
-
FX , two solutions
'
so ,
y
=
X does not define a ( non -
trivial ) function fty
works for a




1- 2 The
graph of a
function

I.
Definition graph :
The
graph of a
function f is the

set of all points IX1Y ) in the
xy
-

plane with X C-
Domf
1×1
and y=f

{ full }
'



graph f- = IX.
y ) EIR
:
XE Domf , y -_

, closed circle : included point
-
2 .




\ excluded
open
circle :
point .




3 Not function
.




every
curve is the
graph of a




The vertical line test :




If any
vertical line intersects the curve more than once then
m u m


not it
the curve is the
graph
mmmm
of a
function , otherwise is


proof :
following the
defining property of a
function ,
only one



element in the is associated with element in the
range
an



domain


Eg
is the
graph of function
-

a




NOT
is the
graph of function
-
a




However . For y 70 ,
the curve is the


graph of the function

Y :[ I , 3] H [o.O ]
domain
range
y= I4-1HT

1. 3 Evan and Odd Functions
I.
Definition Even
function : A function f is even if
f- 1×1 =
f- 1-X ) VIX t Domf

A function
Definition Odd function :
f- is odd if