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ii
133. 6) =Hos (x +571) ana
£9) = log (-r+ vei)
fx) + x)
=log (x+V? 41) + toe(-x
2 £@) pan odd function.
1M 83) = sin(loe(x-+ V7)
fa 2) = sin[ fog (x ex) ]
= sng (Sie 4) “*)
(vies? +3)
1
Vitex +x
= sin log(x+ i? y]
= sia{ og(x+ V+) |
7 ~sin[fog(x+ Vir )
=-f(x)
(2) is an odd function.
= sing
35, f-2)= see|log(—x + i+=F)|
se[loe( -=+Vie)]
see{fog(vi+ xx
|
)]
5]
mle ger
[-! log( Vie +)|
= se[log( VF]
f(x) is an even function.
= sd e{
= sec}
136. Consider option (C),
(a* -1)
fa)= 4
-x(a*=1
w(i-a')_ x02"=l) ey
+a" ated
(2) is an even function.
137. Since, f(x) is even
f(x) = fle)
138. As f{-x)=f%) VxeR
‘. y remains unchanged when x is changed to ~~.
Hence, the graph of y = f(x) is symmetric
about Y-axis.
139. f&) = f- x) > (0 + x) = {0 - x is
symmetrical about x= 0. .
£2 + x) = {(2 —x) is symmetrical about x=
140. Value of the function f(x) does not exist.
= 1or—1ie,, not unique)
x
143.
3S
Gvenesreion= $24]
22a e724
-2(2.4), 8 faa
tea] a [33]
-on 8 [2 ;
5G]
= 66
each term in the summation is one
but less than 2 when i = 33, 34, 35 mes 145.
(46.
1
=1+ 3 [oseaes on (4 >)
~eoe{ 21+) -e(2}]
+ arco 20+ 28)
-eos{2x4)
3
+) cos 2-2 sin 2008 sin{ =
2 2 6
+1 cos ax-asin Ese) 4
2 2 )2
+} (cos2x—00s2x)
0
3
a
3
a
3
ry
3
4
+2=2
=P=2-2
y= log: (2 - 2°) is defined, if2- 2"> 0
= <2
a2t 0, logs x > 0 and x>0
= logsx > 3°= 1,x>4°= 1 andx>0
=x>4'jx>landx>0
=x>4
domain of fis (4, «).
. f(x) is defined, if
1 = logy (2? — 5x + 16) > O and x* = Sx+16>0
=> logio@? — 5x +16) <1 and
2
(5 +32 50 for all real x
2) 4
=> 5x+16<10'=10
=P -5x+6<0
= (r= 3)(&-2)<0
32 0,14 p> Oand.x£0
z
1
= lon (It fe eh + > 0 andx#0
¥ z
a
1
> (ta}> (4) sche >and x #0
= toot, Ay Ht and 40
wy
2 0-I andx #0
domain of f(x) =(0, 1).
Since, e* is defined for all real x.
domain is (— 2, 0)
|. Since, domain of Va? —x* is [-a, a]
domain of V4—x" is [-2, 2].
Since,domain of Vx’ —a” is (=, ~a] U [a, =)
domain of Vx? -16 is (~~,-4] U [4, 0)
1
Since, domain of is (-a, a).
1
domain of is (-3, 3).
1
Since, domain of ————$———= is.
Vee)
(-, a) U (b, «), where a 0
=>xt>(0.5)!
Sat21
xl=1=x=0orl
Domain of the function
jx+2]
= isR-
f(x) 2 ® {2}
|x+¢]
isR- «|
xt
157. f(x) = log (Ve=4 4 o=x)is defined,
when x-4>0and6-x>0
=>x2>4andx<6 = [4, 0) A(-~, 6]
domain of f(x) = [4, 6] — =
158. Dr = fF ER: 1-x>0}nfreR
~&ERIX< A Ger:
ieys~lorxet
= Ee, IV (00, Uli eo oe
bse
Comune
=(,-1] “ett
159. Dr=Dp Dy
. 1
where g(x) = Toga(iza othe) = Vex
Now, Dp= { € R: 1—x>0, logo (1 —2)0}
={xeRix0}
= {xe Rix>2}
Dr= [(-~, 1) - {0}] 9-2, «)
=[-2, 1) - {0}
160. Let g(x) = sin os 32 ana ne) = “ton x)
g(x) is defined, if
-1 3 <1520>x<4
domain ofy=[1, 5] 0 [-,4)
=[1,4)
161.
5
Let 262) =logi-=7 > and
h(x) = fe+5
Now, g(x) is defined, if
D,=(4,5)U 6,2) oi)
h@)
Yx+5 is defined for all real x
Dy=R
From (i) and (ii), we get
domain of f(x) =D, 0 Ds
=RA (4,5) V6,2)}
= (4,5) (6,%)
162. For domain of f(), x + 3 > 0 and
P43 220
=px>—3and (r+ 1) (e+2)#0
= x>-Bandx 4-1, -2
Hence, domain = (-3, ©) ~ {-1,-2}
He hive
0 1 3
3 2
D: +) 3|
x43
164. = |_—,
®” Va-aG-5
f(x) is defined, if
+3) Q-a) (e520
and (2—x) (x5) #0
= (v3) @-2)(e-5) <0 andx #2, 5
domain of ffx) = (20, -3] U 2, 5)
2H 2Oand2- |x| #0
= Uob)(2-b)
(2-h\)
= ([s|-1) ([x]-2) > 0 andx 4-2, 2
= || s Lor fx] >2
2Oandx#-2,2
> -1Sx<1 or(e<—2orx>2)
domain of fix) = [-1, 1] U (0, -2) u (2, 0)
