Engineering Mathematics 1a notes

,Domain Codomain and
range
:
,




codomain


f : IR > o → Rao f-☒
dofnain
"


range
Domain
Range




Composition of functions




f
' '

☐ → E :D → E
:
g




f
f- ( x )
gcx )=T× glflx)) domain off
-9> if = ✗ s to we need to restrict the
compose
-


→




☐
F D
'
,
we need
range of f contained in IR > o so we take domain
E


✗ 2s ( IRIS)


if ✗ Is then
g(f(x))=xFs


Graphs and parametric functions :




2
I
y ✗
= -




•
•




• •


•




of function f.IR IR
parametric
→
every graph a is a function




""
%)
Parametric functions :
•
( cos
'
f :[0,21T] → 1122 ✗ ( t) = cos ( t)



f- (t) ( coslt) , sinlt)) ( t)
yet)= sin
=
•


( COSIO) sin (d)
,




2) ✗ (t) = 1-
It
'




ylt)=t -


Its


-
2 -
I -
213 •
✗ = I -

Yz -12
-
l
'
12 -516 •



0 I 0 •



( 1- ×)
'

516
t = 2
I 112 •


•

2 -
I 213

y2= ( t
'
-
%t3)Z= t
'
-
43 (t )
'
+ Yzg ( TY
}




in terms of
y2 ×

, Inverse Functions :




f :D
-1
if > R inverse of f f : R →D
✗ becomes and vice versa reflected on
y
-



, y=✗

f ✗ 2+1
y
=


s

<

f- ✗
YZ +1
I =




D R




continuous vs .
discontinuous




f- (X ) is continuous at ✗ =a if and
only if
linga f-
1×3=1×11,2 f- ( x) f (a)
=
:
, .




Periodic function :




A function flx) is periodic if we can find a number T such that f( ✗ +T ) =
f( × ) for all values of ✗




Odd and even function :




even : an even function is such that fl -
X)=f(×) for all values of ✗




symmetrical on
y axis




odd :
an odd function is such that fl -
X) = -
flx) for all values of ✗




Properties : odd + odd =
odd even ✗ even = even



odd ✗ odd __ even odd + even =
neither


even + even = even odd ✗ even = odd

, Monday 27th September




✗ = -12 - I
y= -13 -
t



t2=
y2=t°
'
✗ +I
"
-2T + t




ya :( ✗ 1) 21×+15+1×+1)
3- +




point the above satisfies the
a on curve (a) b) is such that ✗
=o,y=b equation