Engineering Mathematics-1

mathematics
Engineering -
I

, Module -
1



Functions
A function f is a rule that
weighs to each element n in a


element called fca )
ut D
exactly one in a set E .




* The domain of the function f is the set of all
possible inputs
of fecal .




The of the function f is the set of all
possible values
*
range
domain
of fecal as a varies
throughout the .




Independent variable ( symbol
A that represent
→
, y = f Ca ) arbitrary number
an the in domain )
a


variable ( that
Dependent A
symbol represent a number
in the
range of f)




x ¥ t
f! s teal

( output)
( input)
function f
machine diagram for a




f- cist
-




fi
- -
→




#
-
.. . - - - - - - - - - -




, :



!
}
rang
the '
!
ffa -
÷
.


s
,
yes ,

!
.


i
I 2 N i
,

i

-
:




domain

, Eixample : find the domain of each function .




⑨ feel ⑨ '
at a
gcais
=



n' -
a


solution The domain of f consists of all values
: ⑨
of a such that at 230
a > -
2

r
'


. The domain is E- 2. n )


⑧ the domain of gcn , is


{ KER n to ,
a # I
}
which can be written in internal notation as


1- A
,
o
) U ( O
, 1) U ( i
,
D)

The vertical line test A the
ay plane
: curve in - is the

of a function of iff vertical line intersects
graph a no


the curve more than once .




Y 't
n
wa
n
c.a.cl

~ is
fan : !
)

I
"


Ca b Ca b I
.

, ,




° ° am

. . this curve can 't
but
represent a function
represent a function
( became function auign two


different values to a ) .

, piecewise defined -
functions
-




The functions which are defined
by different
formulas in different parts of their domains .




Example .

"




{
N it
'




as a
#
' -




il fear .




'
n if a > 2




21

{
I R E C- A. o )
f- Cal =




2 N Elo ,
N )



s) Absolute value function .




{
N it a > o

f- Cnt lay =




-
n if u so




feel = - K C -
0,0 )


flat -
- a o
,
H )
ry
Int
y
-
-




o
7k

, is
4)
I
.




O
'
S n
l




{ If l
*
flat -

-
n


l 2h31
=




-
Even
-
function of odd
-
function
-




A function Fca ) is said to be even if
FC -
al -

-
flag for all n in its domain .




A function Fca ) is said to be odd if
f-C al
-
-

-
-

feng for all u in its domain .




E e) feat = ah - s even function

fl al-
-

-

Ext n' flag =




ii ) fecal = n
'
→ odd function

f- C al
-
=
f- ng? -


n! -


f Ca )

, graph symmetric
~
" If the is
→ Y "


with then
respect to y
- anis

we
say
that the function
is even .




If the graph is
symmetric
about the then
originthe
we


can that function
say
is odd .




Example : Determine whether each of the following
functions is even odd or neither even nor odd
, .
.




att 't -

1) flat = n 2)
glut I -
n 4 Hat 2x - N




solution :
'
i ) t C al -
=
faf -
a = - a - se -
-
-
Latta )

fl -
a) = -
Fca )
i . flu ) is odd function


2) C n' 't
gcn )
offal I i n
-

=
-
-
= - -




gl al
gcn,
-
- -




function
gcn
I . , even

, -


3) h1n1 = ga -
a



ht -
a ) = -
2x -
C - NII -
2x -
RE -
( satay

ht - n ) th Ca ) or ht self
- - heal


.
'




.
hla ) is neither even nor odd .




and function
Increasing decreeing
A function f called interval
increasing
is on an


I if
f- Cm ) s the 4 whenever Ncaa in I


function f called interval
A

I if
is
decreasing on an




f-Cm ) > flxz) whenever me na in I



The function flat = are

→ y
=p
- is
decreasingand on the

interval C -
H
,
)
o


the Eo )
increasing on A
.

, polynomials :




A function play is called a
polynomial if
'
ant an an
-




plat = an . , t .
. - - -

ta , n t Ao


and numbers
where n is non -




negative integer
constant called the coefficient
are ,
a. .
. .


an are
.




of polynomial peal .




any polynomial
* For the domain is A- twig
* If the
leading co -
efficient an to then the

degree of the
polynomial is n .




't
Ex : .


peak 3kt -

2N tf n 't 2
this is the
polynomial of degree 5 .




* A
polynomial tb
of degree
whie is called
of the form i
linear function
is


play an
-
-




*
polynomial
A of degree 2
,

function
plain -
an 't bat e


and called is
quadratic
power function
Attu of the form flat = sea where a
,




power function
is a constant is called a .




"

ri y a
-


n
-



y y


y
y
- .
- -

,I

, Transformation of functions .




Vertical and
Horizontal shifts
suppose c > o .




distance
of
y teal shifts the
graph y feat a
-

te
-
- -




,


c units
upward .




of distance
y teal shifts the
graph y feat a
- -
- -
c -




,


c units downward .




of distance
y ten ) shifts the
graph y feat a
-
-
- - c -


,



units the
c to
right
of distance
y ten + " shifts the
graph y feat a
- -
- -


,



e units to the left