INVERSE TRIGONOMETRY FUNCTIONS EXERCISE–I Q 1. (i) Q.3 Q 4.
Q5.
, (ii) 1, (iii)
(vi)
{7/3, 25/9}
(i)
D : x ε R R : [π /4 , π)
(ii)
D: x ∈
(iii)
D: x∈R
2
(a) x =
π 3
(b) x = 3
or 3 (f) x = Q.14
57
π R : 0 , 2 Q 8.
1 2
, (vi) α
(iv)
D: x∈R
Q.11
π
, n ∈ I}
π − 2
π π
R : − , 2 4
,−
(d) x =
(g) x = Q.20
4 2 9π Q5. 6 cos2x – , so a = 6, b = – 2
Q 6.
(a)
Q 7.
(a) x = n²
(b)
(d) arc tan (x + n) − arc tan x (e)
(c) arc cot
−n+1
Q 9. K = 2 ; cos
k=
, (v)
π4a12π32πn− 3+b 5 π 5 2 9 1 1 −17 1 −±3π2 ,3,17 2 EXERCISE–II π n 126 2 4 366 310 5 +42 an7nb1π+ 2 3 3 2
−π
Q 12.
π 2π R: , 3 3
n∈I ;
(c) x = 0 ,
,y=1
{xx = 2n π +
(viii)
Q 19. x = 1 ; y = 2 & x = 2 ; y = 7
53
, (iv)
(iii) 1 < x < 4 (v) (3/2 , 2]
π − x x = nπ + 4
, (ii) −1, (iii)
Q 2. (i)
(−2, 2) − {−1, 0, 1}
(vii)
, 1
(e) x =
Q 4.
, (vi)
(–∞, sec 2) ∪ [1, ∞) −1/3 ≤ x ≤ 1 (ii) {1, −1} x ∈(−1/2 , 1/2), x ≠ 0
3
Q.13
, (v)
(d) (i) (iv)
Q 6.
Q.12
, (iv)
,1 & cos
11
4
, −1
Q 10. 720
X = Y= 3 − a 2
Q.11
Q 14. (a) (cot 2 , ∞) ∪ (− ∞ , cot 3) (b)
4
4
Q 8. (α2 + β2) (α + β)
or x = n (b) x = ab (c) x =
π2
π
2
G H2
, 1 (c)
∪ − 1 , −
2
2
1 Q15. tan , cot 1 2
Q16. C1 is a bijective function, C2 is many to many correspondence, hence it is not a function
Q17. [eπ /6 , eπ]
Q 18.(a) D : [0, 1] , R : [0, π /2] (b)
Q.19
3π
Q.20
4
1
1
2
2
− ≤x≤
(c) D : [− 1, 1] , R : [0, 2]
x ∈ (–1, 1)
EXERCISE–III Q.1 C
Q.2
B ansal C lasses
π
Q.3 x ∈{− 1, 0, 1}
Q.4 x =
Q.5 B
Functions & Trig.-φ φ -- IV
Q.7 D
Q.8 A
[28]
Q.13
Q.14
Q.15 (a) Q.17
ex − e −x
;
2
(b)
; (c)
1 2
{–1, 1}
Q.16
log
x =1
(i) period of fog is π , period of gof is 2π ; (ii) range of fog is [−1 , 1] , range of gof is [−tan1, tan1]
Q.18 (a) π /2 (b) π (c) π /2 Q.20 ± 1, ± 3, ± 5, ± 15
(d) 70 π
EXERCISE–II Q 1. f −1(1) = y Q.2 (a) – 3/4, (b) 64, (c) 30, (d) 102, (e) 5050 Q.3
(a)
Q 4.
b can be any real number except
Q.6
6016
, (b) 1, (c) [0, 4), (d) – 5 15
Q5. f (x) = 1 – x 2, D = x ∈ R ; range =(– ∞, 1] 4 Q 9. f (x) f (x) = 2 x2 x +1 ,
Q 11. fog (x) =
fof (x) fof (x) =
Q 12. −
3
x
+1 2
4
,
,
≤x≤1 3≤ x≤ 4
−x
,
1−
3
0
2
∪
≤x<1 3− x , 1 ≤ x ≤ 2 ; gof (x) gof (x) = ; x −1 , 2 < x ≤ 3 5−x , 3 < x ≤ 4 1 1≤ x ≤ 0 − (+1log +xx2)x , 1 −if 0< x≤1 2 − x1002 1 ,≤x x0 ≤< 0x ≤ 2 1log x −− 2,x 1x − ; gog (x) = x g( x,) = 0 < x ≤ 2 2 4 − x , 2 < x x ≤if 3x >1
3 − 1 3 + 1 , 2 2
Q.14
1002.5
Q.15
5049
Q.17
20
Q 18. (0 , 1)
0
Q.13 Q.16
x = 0 or 5/3
g (x) = 3 + 5 sin(nπ + 2x – 4), n ∈ I
∪ {1, 2, ....., 12} ∪ (12, 13)
Q 19. f (x) f (x) = sin x + x −
π 3
EXERCISE–III Q.1 (hofog)(x) = h(x2) = x2 for x ∈ R , Hence h is not not an identity function , fog is not invertible Q.2 (a) A, (b) B Q.3 (fog)(x) (fog) (x) = e3x − 2 ; (gof) (x) = 3 ex − 2 ; Domain of (fog)–1 = range of fog = (0, ∞); Domain of (gof)–1 = range of gof = ( − 2, ∞) Q.4 B Q.5 D Q.6 {(1, 1), (2, 3), (3, 4), (4, 2)} ; {(1, 1), (2, 4), (3, 2), (4, 3)} and {(1, 1), (2, 4), 4), (3, 3), (4, 2)} Q.7 (a) B, (b) A, (c) D, (d) A, (e) D Q.8 (a) D ; (b) A Q.9 (a) D , (b) A Q.10 C Q.11 (a) A ; (b) D
B ansal C lasses
Functions & Trig.-φ φ -- IV
[27]
ANSWER KEY FUNCTIONS
EXERCISE–I
∪
Q 1. (i)
π π − 4 , 4
(vii) (−1 < x < −1/2) U (x > 1)
1 − 5 (viii) , 0 2
1 + 5
∪
∪ [ 3,4) 2Kπ < x < (2K + 1)π
(iii) (– ∞ , – 3]
, ∞ (ix) (−3, −1] U {0} U [ 1,3 )
2
1 π 5π , ∪ ,6 6 3 3
(xii) −
(xi) (0 , 1/4) U (3/4 , 1) U {x : x ∈ N, x ≥ 2}
(xiii) [– 3,– 2) (xv)
∪ (2, ∞)
1 1 1 0 , ∪ , (v) (3 − 2π < x < 3 − π) U (3 < x ≤ 4) (vi) 100 100 10
(iv) (– ∞, – 1) ∪ [0, ∞)
(x) { 4 } ∪ [ 5, ∞ )
∪
1 3π 5 π − − , 4 , (ii) 4 4 2
(xiv)
φ
but x ≠ 1 where K is non−negative integer
(xvi) {x 1000 ≤ x < 10000} (xvii) (–2, –1) U (–1, 0) U (1, 2)
(xviii) (1, 2) ∪
(xix) (− ∞ , −3) ∪ (−3 , 1] ∪ [4 , ∞) Q 2. (i) D : x ε R
(ii) D = R ; range [ –1 , 1 ]
R : [0 , 2]
(iii)
D : {xx ∈ R ; x ≠ −3 ; x ≠ 2} R : {f(x)f(x) ∈R , f(x) ≠ 1/5 ; f(x) ≠ 1}
(iv)
D : R ; R : (–1, 1)
(vi)
D : x ∈ (2nπ, (2n + 1)π) −
[
]
255π R :− 3 π3, 6 (v) D : −1 ≤ x ≤ −2, , 42 4 2 nπ + π6 , 2 nπ + π2 , 2 nπ + 56π , n ∈ I and
{
}
R : loga 2 ; a ∈ (0, ∞) − {1} ⇒ Range is (–∞, ∞) – {0}
1 1 1 ∪ , 6 6 3
(vii)
D : [– 4, ∞) – {5}; R : 0,
Q.4
(a) neither surjective nor injective
(b) surjective but not injective
(c) neither injective nor surjective Q.5 f 3n(x) = x ; Domain = R − {0 , 1} Q.6 Q.7 (a) 2Kπ ≤ x ≤ 2Kπ + π where K ∈ I (b) [−3/2 , −1] 1 Q.8 (i) (a) odd, (b) even, (c) neither odd nor even, (d) odd, (e) neither odd nor even, (f) even,
(g) even, Q.9 Q.10 Q.12
(h) even;
(ii)
−1 + 2
5
,
−1 − 2
5
,
−3 + 2
5
,
−3 −
5
2
(a) y = log (10 − 10x) , − ∞ < x < 1 (b) y = x/3 when − ∞ < x < 0 & y = x when 0 ≤ x < + ∞ f −1(x) = (a − xn)1/n (a) f(x) = 1 for x < −1 & −x for −1 ≤ x ≤ 0; (b) f(x) = −1 for x < −1 and x for
B ansal C lasses
φ Functions & Trig.-φ φ - IV
−1 ≤ x ≤ 0
[26]
EXERCISE–III Q.1
Q.2
The number of real solutions of tan−1
+ sin−1 x 2 + x + 1 =
(A) zero
(D) infinite
(B) one
(C) two
π 2
is :
[JEE '99, 2 (out of 200)]
Using the principal values, express the following as a single angle : 142 1 + 2 tan−1 + sin−1 . 5 65 5
3 tan−1
Q.3
Solve, sin−1
+ sin−1
Q.4
Solve the equation:
[ REE '99, 6 ]
= sin−1x, where a2 + b2 = c2, c ≠ 0.
[REE 2000(Mains), 3 out of 100]
[ REE 2001 (Mains), 3 out of 100] Q.5
+ cos–1 x 2
If sin–1 (A) 1/2
Q.6
Q.7
Prove that cos tan
(B) 1
–1
sin cot
1 1 , 2 2
x4 2
+
x6 4
π
2
− ........ =
for 0 < | x | <
–1
x=
+1 x2 + 2 x2
is
1 3 (B) − , 4 4
then x equals to
[JEE 2001(screening)] (D) – 1
(C) – 1/2
Domain of f (x) = (A) −
−
[JEE 2002 (mains) 5]
ab1x 2 −1 x−1x 2 x 3 2 cos 6 x + cos 3 3 x x ( x 1 ) + π 1 − ccxsin − (2+x ) +− ........ 4 6 2 2
(
)
1 1 (C) − , 4 4
(
π
)= 2
1 1 (D) − , 4 2 [JEE 2003 (Screening) 3]
Q.8
If sin cot −1 ( x + 1) (A) –
1 2
B ansal C lasses
= cos(tan −1 x ) , then x = (B)
1 2
(C) 0
φ Functions & Trig.-φ φ - IV
(D)
9
4 [JEE 2004 (Screening)]
[25]
is equal to
120π
Q.10
If the value of
Q.11
If X = cosec . tan−1 . cos . cot−1 . sec . sin−1 a & Y = sec cot −1 sin tan−1 cosec cos−1 a ; where 0 ≤ a ≤ 1 . Find the relation between X & Y . Express them in terms of ‘a’.
Q.12
Find all values of k for which there is a triangle whose angles have measure tan–1
k
, find the value of k.
1 + k , 2
, tan–1
1 + 2k . 2
and tan–1
Q.13
Prove that the equation ,(sin−1x)3 + (cos−1x)3 = α π3 has no roots for α <
Q.14
Solve the following inequalities : (a) arc cot2 x − 5 arc cot x + 6 > 0
Q.15
Q.16
Consider the two equations in x ; The sets X1, X2
⊆ [−1, 1]
;
and α >
4 arc cotx – arc cot 2 x – 3 > 0
(i) sin Y1, Y2
32
(c) tan2 (arc sin x) > 1
(b) arc sin x > arc cos x
Solve the following system of inequations 4 arc tan2x – 8arc tanx + 3 < 0 &
1
sin −1 x =0 (ii) cos y
=1
⊆ I − {0}
are such that
X1 : the solution set of equation (i) X2 : the solution set of equation (ii) 7cos 1 −1 xn (i) possess Y1 : the set of all integral values of y for which equation a solution cos −1 1 + ( k − 1) k (k + 1)(k + 2) 8 Lim Y2 : the set of all integral values of y for which equation a solution 2 ∞y (ii) possess n→ k ( k + 1) k 2 = Let : C1 be the correspondence : X 1 → Y1 such that x C 1 y for x ∈ X1 , y ∈ Y1 & (x , y) satisfy (i).
∑
C2 be the correspondence : X2 → Y2 such that x C 2 y for x ∈ X2 , y ∈ Y2 & (x, y) satisfy (ii). State with reasons if C1 & C2 are functions ? If yes, state whether they are bijjective or into? cos −1 sin x
( (
+ π3 ) )
4 − 2 cosx , g(x) = cosec −1 & the function h(x) = f(x) 3 defined only for those values of x, which are common to the domains of the f unctions f(x) & g(x). Calculate the range of the function h(x).
Q.17
Given the functions f(x) = e
Q.18
(a)
2 2x −1 1 − x are identical functions, then compute If the functions f(x) = sin−1 & g(x) = cos 2 2
1+ x
1+ x
their domain & range . (b)
Q.19
If the functions f(x) = sin−1 (3x − 4x3) & g(x) = 3 sin −1 x are equal functions, then compute the maximum range of x.
, , Ð1
( )+
Ð1
( )+
( Ð 2)(3 Ð 7) = 2,
.
Ð1
( ).
2x 2 + 4 Q.20 Solve for x : sin sin 2 < π – 3. 1 + x –1
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[24]
EXERCISE–II Q.1
a π 1 − cos−1 = 2 b b a 4 2
Prove that: (a) tan
+ tan
a−b x b + a cos x . tan = cos−1 (c) 2 tan−1 2 a + b a + b cos x
y x = 2 tan−1 tan . tan
(b) cos−1
2
2
Q.2
2 2 1 1+ x − 1− x − If y = tan prove that x² = sin 2y. 1 + x2 + 1 − x2
Q.3
If u = cot−1 cos2 θ
Q.4
1 + x If α = 2 arc tan & 1 − x
− tan−1
cos2 θ then prove that sin u = tan2 θ.
1 − x 2 β = arc sin 1 + x 2 for 0 < x < 1 , then prove that
α + β = π, what the
value of α + β will be if x > 1.
Q.5
1 If x ∈ −1,− then express the function f (x) = sin –1 (3x – 4x3) + cos–1 (4x3 – 3x) in the form of 2
acos–1 x + bπ , where a and b are rational numbers. Q.6
Find the sum of the series: 1 2 −1 sin−1 + sin−1 + ..... + sin−1
(a)
2
2 n −1
−1 1
−
n
−1
(c)
tan + tan−1 + ..... + tan −1 1 + 22 n − 1 + ..... ∞ 3 cot−17 + cot−113 + cot−121 + cot−131 + ...... to n terms.
(d)
tan−1
(e)
tan−1
(b)
Q.7
6
xn
1cos 23 2 1 x π2x3− −1x1111+ cos−1y+ a ...... ∞ β + cos n ( n 1 ) + b 18 1a936 32 17 3 x12x+ 3 13 5 7 yb x +1 cos 2x824x++++cos
+ tan−1 + tan−1
+ tan−1
+ tan−1 + tan−1
+ tan−1
to n terms.
+ ..... ∞
Solve the following (a) cot−1x + cot−1 (n² − x + 1) = cot −1 (n − 1) (b) sec−1 − sec−1 = sec−1b − sec−1a a ≥ 1; b ≥ 1, a ≠ b. (c) tan−1
+ tan−1
= tan−1
1 −1 α 1 −1 β α3 2 cosec tan + sec 2 tan β as an integral polynomial in 2 α 2 2
Q.8
Express
Q.9
Find the integral values of K for which the system of equations ;
α & β.
K π2 2 + = arc cos x ( arc sin y ) 4 possesses solutions & find those solutions . 4 π (arc sin y)2 . (arc cos x) = 16
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[23]
Q.10
If arc sinx + arc siny + arc sinz = π then prove that :
(x, y, z > 0)
(a) (b) x4 + y4 + z4 + 4 x2y2z2 = 2 (x2 y2 + y2 z2 + z2x2)
ab + 1 bc + 1 ca + 1 + cot–1 + cot–1 . a − b b − c c − a
Q.11
If a > b > c > 0 then find the value of : cot–1
Q.12
Solve the following equations / system of equations:
π (a) sin−1x + sin−1 2x =
(b) tan−1
(c) tan−1(x−1) + tan−1(x) + tan−1(x+1) = tan−1(3x)
1 π (d) sin−1 + cos−1x =
3
(e) cos−1
2x + tan−1 2 x
−1
(g) 2 tan−1x = cos−1 Q.13
2π 3
− cos−1
5
1 1 + 4x
2 = tan−1 2 x
4
2π (f) sin−1x + sin−1y = & cos−1x − cos−1y = 3
(a > 0, b > 0).
Let l1 be the line 4x + 3y = 3 and l2 be the line y = 8x. L 1 is the line formed by reflecting l1 across the line y = x and L 2 is the line formed by reflecting l2 across the x-axis. If θ is the acute angle between L1 and L2 such that tanθ=
Q.14
=
+ tan−1
, where a and b are coprime then find (a + b).
Let y = sin–1(sin 8) – tan–1(tan 10) + cos–1(cos 12) – sec–1(sec 9) + cot –1(cot 6) – cosec–1(cosec 7). axπ2− 1−abπ21π 2 π 1∈ If y simplifies to aπ + b then find (a – b). − y 2 +−1z 1 − 46 x x , +33 yπ 1 z2 π =2 +xyztan−1 − tan 13 π 036 ,1−−21 −2x ∈ sin 3 + sin cos cos 2 1b a b + x 85 7 8 + 21 2 2 7
Q.15
Show that :
Q.16
Let α = sin–1 (i)
=
+ cot −1 c
13π 7
4 8 , β = cos–1 and γ = tan–1 , find (α + β + γ ) and hence prove that 5 15
∑ cot α = ∏ cot α ,
∑ tan α · tan β = 1
(ii)
Q.17
Prove that : sin cot–1 tan cos–1 x = sin cosec –1 cot tan–1x = x
Q.18
If sin2x + sin2y < 1 for all x, y
Q.19
Find all the positive integral solutions of, tan−1x + cos−1
Q.20
Let f (x) = cot–1 (x2 + 4x + α2 – α) be a function defined R →
where x
∈ (0,1]
R then prove that sin–1 (tanx . tany)
y 1 + y2
.
3 = sin−1 . 10
then find the complete set of real
values of α for which f (x) is onto.
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[22]
Q.4
Find the domain of definition the following functions. ( Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f(x) = arc cos
(ii)
(iii) f (x) = (iv) f(x) =
1 − sin x log 5 (1 − 4x ) 2
+ cos −1 (1 − {x}) , where {x} is the fractional part of x . 3 − 2 x 5
(v) f (x) = 3 − x + cos −1
+ log6 ( 2 x − 3) + sin −1 (log 2 x)
3 (vi) f (x) = log10 (1 − log7 (x2 − 5 x + 13)) + cos−1 9π x 2 + sin 2
(vii) f(x) = e
sin −1 ( x2 )
(viii) f(x) =
Q.5
+ tan −1 − 1 + ln ( 2 x
x − [x]
)
cos sin (cos x) + ln (− 2 cos2 x + 3 cos x + 1) + e
−1 2 sin x
+ 1 2 2 sin x
Find the domain and range of the following functions . (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) f (x) = cot−1(2x − x²)
(ii) f (x) = sec−1 (log3 tan x + logtan x 3)
2 x2 + 1 1 − (iii) f(x) = cos x2 + 1
−1 2 2 y22= 2 − + tan log 5 x 8 x 4 (iv) f (x) x ) 3 x2n.−xy 4−1( 1 + −21m xcos y−x x) 2 − sin log ( 4 + (sin x ) sin 10 1b +2− cos α 5 + 2x2 = sin α 2 x − na22b am b
Q.6
Find the solution set of the equation, 3 cos−1 x = sin−1 1 − x2
Q.7
Prove that: (a) sin–1 cos (sin−1 x) + cos –1 sin (cos–1 x) =
π
2 1 1 1 1 − − − − (b) 2 tan (cosec tan x − tan cot x) = tan x
|x|
≤1
(x ≠ 0)
2 p q 2 M N = tan−1 2 where M = mp − nq, N = np + mq, + tan−1 2 2 p − q M − N 2
(c) tan−1
n
,
(4 x 2 − 1) .
<1 ;
q
<1
N
<1 m p M (d) tan (tan−1 x + tan−1 y + tan−1 z) = cot (cot −1 x + cot−1 y + cot−1 z) and
x 1 + 2 2
Q.8
Find the simplest value of, arc cos x + arc cos
Q.9
If cos−1
x a
+ cos−1
B ansal C lasses
3
− 3 x 2 , x ∈ 1 , 1
= α then prove that
φ Functions & Trig.-φ φ - IV
2
.
[21]
11. (a) y = cosec −1 (cosec x),
y = cosec (cosec −1 x) , = x
11. (b)
= x
x ε R − { nπ , n ε I },
x ≥ 1 , y ≥ 1,
y is aperiodic
y is periodic with period 2 π
12. (a) y = sec −1 (sec x) ,
12. (b) y = sec (sec −1 x) ,
= x y is periodic with period 2π ;
π
2
x ∈ R – (2 n − 1)
n
= x x ≥ 1 ; y ≥ 1], y is aperiodic
π ∈ I y ∈ 0 ,
π , π ∪ 2 2
16 π73∈62 +−−−11π1 3,50 ∪ 0−1, π3− 1 7 −1 36 y cot cos 1 − + sin sin + + cos 2 25 633 2 3 5 2 25 13 325 2213 EXERCISE–I
Q.1
Find the following
(i) tan cos
−1
1 2
π
− 1 3
2π (iv) tan−1 tan
Q.2
(v) cos tan −1
3
7π (iii) cos−1 cos 6
3 4
(vi) tan
Find the following : −1 − 3 π + (ii) cos cos 2 6
π − 3 (i) sin − sin −1 2 2
−1
3π 4π (iii) tan−1 tan (iv) cos−1 cos 4 3
3 sin 2α tan α (vi) tan−1 + tan−1 where 5 + 3 cos 2α 4
3
(v) sin cos 5 Q.3
− 1
−1 (ii) sin 3 − sin 2
+ tan −1
−π 2
<α<
Prove that: (a) 2 cos−1
+ cot−1
(c) arc cos
− arc
cos
+
cos−1 =
=π
(b)
=π
π 6
2
(d) Solve the inequality: (arc sec x) – 6(arc sec x) + 8 > 0
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[20]
7. (a) y = sin −1 (sin x) , x ∈ R ,
7.(b)
,
Periodic with period 2 π
y = sin (sin −1 x) ,
= x x ∈ [− 1 , 1] , y ∈ [− 1 , 1] , y is aperiodic
8. (a) y = cos −1(cos x), x ∈ R, y ∈[0, π], periodic with period 2 π
8. (b) y = cos (cos −1 x) ,
= x
= x x ∈ [− 1 , 1] , y ∈ [− 1 , 1], y is aperiodic
9. (a) y = tan (tan −1 x) , x ∈ R , y ∈ R , y is aperiodic =x
9. (b) y = tan −1 (tan x) , = x y
∈ −
π π
2
,
2
π
π π
x ∈ R − (2 n − 1) 2 n ∈ I , y ∈ − 2 , 2 ,
periodic with period π
10. (a) y = cot −1 (cot x) ,
10. (b) y = cot (cot −1 x) ,
= x x ∈ R − {n π} , y ∈ (0 , π) , periodic with
B ansal C lasses
π
= x x ∈ R , y ∈ R , y is aperiodic
φ Functions & Trig.-φ φ - IV
[19]
INVERSE TRIGONOMETRIC FUNCTIONS SOME USEFUL GRAPHS 1.
y = sin − 1 x , x ≤ 1 ,
3.
y = tan −1 x , x ∈ R , y ∈ − 2 , 2
π π
2.
y = cos −1 x , x ≤ 1 , y ∈ [0 , π]
4.
y = cot −1 x , x ∈ R , y ∈ (0 , π)
π , π 2 2
y ∈ −
5.
y = sec −1 x , x ≥ 1 , y
B ansal C lasses
∈ 0 , π ∪ π , π 2 2
π 6. y = cosec −1 x , x ≥ 1 , y ∈ − 2 , 0 ∪
φ Functions & Trig.-φ φ - IV
π 0 , 2
[18]
P−5
tan−1 x + tan−1 y = tan−1
where x > 0 , y > 0 & xy < 1
= π + tan−1
+y where x > 0 , y > 0 & xy > 1 1 − xy x
x−y tan−1 x − tan−1y = tan−1
1 + xy
P−6
(i)
2 2 sin−1 x + sin−1 y = sin−1 x 1 − y + y 1 − x
Note that : x2 + y2 ≤ 1 (ii)
π 2
where x ≥ 0 , y ≥ 0 & x2 + y2 > 1
π
⇒
(iii)
sin–1x – sin–1y =
(iv)
cos−1 x + cos −1 y = cos −1 x y
2
< sin−1 x + sin−1 y < π where x > 0 , y > 0 1− x 2 1− y 2
m
where x ≥ 0 , y ≥ 0
x + y + z − xyz If tan−1 x + tan−1 y + tan−1 z = tan−1 1 − x y − y z − z x if, x > 0, y > 0, z > 0 & xy + yz + zx < 1 1 1 1 − − − Note : (i) If tan x + tan y + tan z = π then x + y + z = xyz
π If tan−1 x + tan−1 y + tan−1 z = 1πx2+x−then xy + yz + zx = 1 x2 11y −xy 21+− yy 2 1−−yx 21− x 2 sin 2 3 2
(ii) P−8
where x ≥ 0 , y ≥ 0 & (x2 + y2) ≤ 1
0 ≤ sin−1 x + sin−1 y ≤
⇒
sin−1 x + sin−1 y = π − sin−1 Note that : x2 + y2 >1
P−7
where x > 0 , y > 0
2 tan−1 x = sin−1
= cos −1
1− x
2
1+ x
2
1 + − xy
2x = tan−1 2 1− x
Note very carefully that :
2 tan −1 x 2x π − 2 tan −1 x sin−1 = 2 1+ x − π + 2 tan −1 x) (
2x tan−1 2 1− x
≤1 x>1 x < −1
if
x
if if
2 2 tan −1 x if x ≥ 0 11− x − cos = −1 1 + x2 − 2 tan x if x < 0
2tan −1 x if x < 1 = π+ 2tan −1 x if x < − 1 −(π−2tan −1 x ) if x > 1
REMEMBER THAT : (i)
3π sin−1 x + sin−1 y + sin−1 z =
(ii)
cos−1 x + cos−1 y + cos−1 z = 3π
⇒ x=y=z=1 ⇒ x = y = z = −1
(iii)
tan−1 1 + tan−1 2 + tan−1 3 = π
and
2
B ansal C lasses
tan−1 1 + tan−1 12 + tan−1
φ Functions & Trig.-φ φ - IV
=
[17]
KEY CONCEPTS (INVERSE TRIGONOMETRY FUNCTION) GENERAL DEFINITION(S): 1.
sin−1 x , cos−1 x , tan−1 x etc. denote angles or real numbers whose sine is x , whose cosine is x and whose tangent is x, provided that the answers given are numerically smallest available . These are also written as arc sinx , arc cosx etc . If there are two angles one positive & the other negative having same numerical value, then positive angle should be taken .
2.
PRINCIPAL VALUES AND DOMAINS OF INVERSE CIRCULAR FUNCTIONS : (i)
y = sin−1 x where
(ii)
y = cos−1 x where
(iii)
y = tan−1 x where x ∈ R ; −
(iv)
y = cosec −1 x where x ≤ − 1 or x ≥ 1 ; −
(v)
y = sec−1 x where x ≤ −1 or x ≥ 1 ; 0 ≤ y ≤ π ; y ≠
(vi)
y = cot−1 x where x ∈ R , 0 < y < π and cot y = x .
NOTE THAT : (a) (b)
P−1
P−2
;
−1 ≤ x ≤ 1
and sin y = x . ; 0 ≤ y ≤ π and cos y = x .
π 2
π and tan y = x . 2
π 2
≤y≤
π , y ≠ 0 and cosec y = x . 2
π 2
and sec y = x .
1st quadrant is common to all the inverse functions . 3rd quadrant is not used in inverse functions .
(c) 3.
−1 ≤ x ≤ 1
4th quadrant is used in the
CLOCKWISE DIRECTION
i.e.
π1 πFUNCTIONS PROPERTIES OF INVERSE CIRCULAR : π −2 ≤ y ≤ 0 x(ii) 2 cos (cos 2 −1 x) = x , (i) sin (sin−1 x) = x , −1 ≤ x ≤ 1
.
−1 ≤ x ≤ 1
(iii) tan (tan−1 x) = x , x ∈ R
(iv) sin−1 (sin x) = x , − π ≤ x ≤ π
(v) cos−1 (cos x) = x ; 0 ≤ x ≤ π
(vi) tan−1 (tan x) = x ; −
(i) cosec −1 x = sin−1
1 x
;
(iii) cot−1 x = tan−1
;
= π + tan−1
π 2
2
π 2
x ≤ −1 , x ≥ 1
;
(ii) sec−1 x = cos −1
2
x ≤ −1 , x ≥ 1 x>0 ;
x<0
sin−1 (−x) = − sin−1 x , −1 ≤ x ≤ 1 tan−1 (−x) = − tan−1 x , x∈R 1 1 − − cos (−x) = π − cos x , −1 ≤ x ≤ 1 cot−1 (−x) = π − cot−1 x , x ∈ R
P−3
(i) (ii) (iii) (iv)
P−4
(i) sin−1 x + cos−1 x = (iii) cosec−1 x + sec −1 x =
B ansal C lasses
−1 ≤ x ≤ 1
(ii) tan−1 x + cot−1 x =
x∈R
x ≥ 1 φ Functions & Trig.-φ φ - IV
[16]
(c)
The domain of definition of f (x) = (A) R \ {− 1, − 2}
is :
(B) (− 2, ∞)
(C) R \ {− 1, − 2, − 3}
(D) (− 3, ∞) \ {− 1, − 2}
(d)
Let E = {1, 2, 3, 4 } & F = {1, 2}. Then the number of onto functions from E to F is (A) 14 (B) 16 (C) 12 (D) 8
(e)
Let f (x) =
, x
(A)
≠ − 1 . Then for what value of α is f (f (x)) = x ? (B)
−
(D) − 1. [ JEE 2001 (Screening) 5 × 1 = 5 ]
(C) 1
2
Q.8(a) Suppose f(x) = (x + 1) 2 for x > –1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y = x, then g(x) equals (A) – x – 1, x > 0
(B)
1 (x + 1) 2
, x > –1
x + 1 , x > –1
(C)
(D)
x – 1, x > 0
(b) Let function f : R → R be defined by f (x) = 2x + sinx for x ∈ R. Then f is (A) one to one and onto (B) one to one but NOT onto (C) onto but NOT one to one (D) neither one to one nor onto [JEE 2002 (Screening), 3 + 3] 2 x +x+2 Q.9(a) Range of the function f (x) = 2 is x + x +1 7 (A) [1, 2] (B) [1, ∞ ) (C) (D) 1, 3 x (b) Let f (x) = defined from (0, ∞ ) → [ 0, ∞ ) then by f (x) is 1+ x α x−271one(A) one- one but not onto (B) (x + 3one ) and onto log f 2f (b) = b 22 , 3x + 2one and onto xx(D) (C) Many one but not onto [JEE 2003 (Scr),3+3] ++31Many Q.10 Let f (x) = sin x + cos x, g (x) = x 2 – 1. Thus g ( f (x) ) is invertible for x ∈
π
(A) − , 0 2
π
(B) − , π 2
π π
(C) − , 4 4
Q.11(a) If the functions f (x) and g (x) are defined on R → R such that 0,
f (x) = x,
x ∈ rational x ∈ irrational
then (f – g)(x) is (A) one-one and onto (C) one-one but not onto
, g (x) =
0,
x ∈ irrational
x,
x ∈ rational
(D) 0,
π 2
[JEE 2004 (Screening)]
(B) neither one-one nor onto (D) onto but not one-one
(b) X and Y are two sets and f : X → Y. If {f (c) = y; c ⊂ X, y ⊂ Y} and {f –1(d) = x; d ⊂ Y, x ⊂ X}, then the true statement is (A) (B) f −1 (f (a ) ) = a
(
(C) f f −1 (b)
B ansal C lasses
)= b , b ⊂ y
(D) f −1 (f (a ) ) = a , a ⊂ x
φ Functions & Trig.-φ φ - IV
[JEE 2005 (Scr.)]
[15]
Q.18
Find the set of real x for which the function f(x) =
is not defined, where [x]
denotes the greatest integer function. Q.19
A is a point on the circumference of a circle. Chords AB and AC divide the area of the circle into three equal parts . If the angle BAC is the root of the equation, f (x) = 0 then find f (x) .
Q.20
If for all real values of u & v, 2 f(u) cosv = f(u + v) + f(u − v), prove that, for all real values of x (i) f(x) + f(− x) = 2a cos x (ii) f(π − x) + f(− x) = 0 (iii) f(π − x) + f(x) = − 2b sin x . Deduce that f(x) = a cosx − b sin x, a, b are arbitrary constants.
EXERCISE–III Q.1
If the functions f , g , h are defined from the set of real numbers R to R such that ; f (x)= x − 1, g(x) = x 2
2
0, if x ≤ 0 ; then find the composite function ho(fog) & determine + 1 , h(x) = x, if x ≥ 0
whether the function (fog) is invertible & the function h is the identity function. Q.2(a) If g (f(x)) = sinx & f (g(x)) = (A) f(x) = sin2 x , g(x) =
[REE '97, 6]
, then : (B) f(x) = sinx, g(x) = x
x
(C) f(x) = x2 , g(x) = sin x
(D) f & g cannot be determined
(b) If f(x) = 3x − 5, then f −1(x) (A) is given by
1 3x
(B) is given by
−5
x +5 3
Q.3
x (x −212) not (C) does not exist because f is not one −one (D) +2 xdoes −2xx4 − 41 exist because f is not onto 1x− xsin [JEE'98, 2 + 2] 1x+−x 1 + 12 − x − 11 [ 2 2] [ ] If the functions f & g are defined from the set of real numbers R to R such that f(x) = ex, g(x) = 3x − 2, then find functions fog & gof. Also find the domains of functions (fog)−1 & (gof)−1. [ REE '98, 6 ]
Q.4
If the function f : [1, ∞) → [1, ∞) is defined by f(x) = 2x (x − 1), then f −1(x) is :
(
(A)
(B)
1
(1 + 2
1 + 4 log2 x
)
(C)
1 2
)
(1 −
1 + 4 log2 x
)
[ JEE '99, 2 ]
(D) not defined
Q.5
The domain of definition of the function, y (x) given by the equation, 2x + 2y = 2 is : (A) 0 < x ≤ 1 (B) 0 ≤ x ≤ 1 (C) − ∞ < x ≤ 0 (D) − ∞ < x < 1 [ JEE 2000 (Screening), 1 out of 35 ]
Q.6
Given x = {1, 2, 3, 4}, find all one−one, onto mappings, f : X → X such that, f (1) = 1 , f (2) ≠ 2 and f (4) ≠ 4 . [ REE 2000, 3 out of 100 ]
− 1 , x < 0 Q.7(a) Let g(x) = 1 + x − [ x ] & f (x) = 0 , x = 0 . Then for all x , f (g (x)) is equal to 1 , x>0 (A) x (B) 1 (C) f (x) (D) g (x) (b) If f : [1,
∞) →
(A)
B ansal C lasses
[2 ,
∞)
is given by , f (x) = x + (B)
1 x
, then f −1 (x) equals
(C)
φ Functions & Trig.-φ φ - IV
(D) 1
− [14]
Q.5
A function f : R → R satisfies the condition, x2 f (x) + f (1 – x) = 2x – x4 . Find f (x) and its domain and range.
Q.6
Suppose p(x) is a polynomial with integer coefficients. The remainder when p(x) is divided by x – 1 is 1 and the remainder when p(x) is divided by x – 4 is 10. If r (x) is the remainder when p(x) is divided by (x – 1)(x – 4), find the value of r (2006).
Q.7
Prove that the function defined as , f (x) = f (x) is odd as well as even. ( where {x} denotes the fractional part function )
Q.8
1 x
2 f(x) + xf
In a function
(i) f(2) + f(1/2) = 1
Prove that Q.9
πx 1 + = 4 cos2 + x cos 2 4 and (ii) f(2) + f(1) = 0
2 sin π x
A function f , defined for all x , y ∈ R is such that f (1) = 2 ; f (2) = 8 & f (x + y) − k xy = f (x) + 2y2 , where k is some constant . Find f (x) & show that : = k for x + y ≠ 0.
f (x + y) f Q.10
− 2f
Let ‘f’ be a real valued function defined for all real numbers x such that for some positive constant ‘a’ the 1
2
equation f (x + a ) = + f (x) − ( f (x)) holds for all x . Prove that the function f is periodic . 2
Q.11
Q.12
Q.13
f (x) = −1 + x − 2 , 0 ≤ x ≤ 4 1 π 1 g (x) = 2 − x , − 1 ≤ x ≤ 3 − | ln{ x }| | ln{ x }| + ythe − {xof xxeof graphs } fog (x) where ever. it exists Then find fog (x) & gof (x) . Draw rough sketch & gof (x)
If
defined by the implicit equation, Find the domain of definition of the implicit function {x} otherwise , then 4 x2 − 1 y x4 . 3 + 2 = 2
Let {x} & [x] denote the fractional and integral part of a real number x respectively. Solve 4{x}= x + [x] 9x
1 2 3 2005 + f + f + ....+ f 2006 2006 2006 2006
then find the value of the sum f
Q.14
Let f (x) =
Q.15
Let f (x) = (x + 1)(x + 2)(x + 3)(x + 4) + 5 where x [a, b] where a, b ∈ N then find the value of (a + b).
Q.16
Find a formula for a function g (x) satisfying the following conditions (a) domain of g is (– ∞, ∞) (b) range of g i s [–2, 8] (c) g has a period π and (d) g (2) = 3
Q.17
9x
+3
∈ [–6, 6]. If the range of the function is
3 4 The set of real values of 'x' satisfying the equality + = 5 (where [ ] denotes the greatest integer x x b b function) belongs to the interval a , where a, b, c ∈ N and is in its lowest form. Find the value of c c a + b + c + abc.
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[13]
→
3
, ∞ defined as, f(x) = x2 − x + 1. Then solve the equation f (x) = f −1 (x).
Q.16
A function f :
Q.17
1 Function f & g are defined by f(x) = sin x, x∈R ; g(x) = tan x , x∈R − K + π
4
where K ∈ I . Find Q.18
(i) periods of fog & gof.
Find the period for each of the following functions : (a) f(x)= sin4x + cos4x (b) f(x) = cosx (d) f(x)= cos
x − sin
2
(ii) range of the function fog & gof .
(c) f(x)= sinx+cosx
x.
Q.19
Prove that the functions ; (c) f(x) = x + sinx
(a) f(x) = cos (d) f(x) = cos x2
(b) f(x) = sin x are not periodic .
Q.20
Find out for what integral values of n the number 3π is a period of the function : f(x) = cos nx . sin (5/n) x.
EXERCISE–II
Q.1
Let f be a one−one function with domain {x,y,z} and range {1,2,3}. It is given that exactly one of the following statements is true and the remaining two are false . f(x) = 1 ; f(y) ≠ 1 ; f(z) ≠ 2 . Determine f− 1(1)
Q.2
Solve the following problems from (a) to (e) on functional equation.
(a)
The function f (x) defined on the real numbers has the property that f ( f ( x ))· (1 + f ( x )) = – f (x) for all x in the domain of f . If the number 3 is in the domain and range of f, compute the value of f (3).
(b)
Suppose f is a real function satisfying f (x + f (x)) = 4 f (x) and f (1) = 4. Find the value of f (21). 22 2 3 f 1x( y) Let f ' ' be a function defined from R + → R+ . If [ f (5 (xy)] )= x 7542 , ∞ f (2) = 6, find the value of f (50).
(c)
for all positive numbers x and y and
(d)
Let f (x) be a function with two properties (i) for any two real number x and y, f (x + y) = x + f (y) and (ii) f (0) = 2. Find the value of f (100).
(e)
Let f be a function such that f (3) = 1 and f (3x) = x + f (3x – 3) for all x. Then find the value of f (300).
Q.3(a) A function f is defined for all positive integers and satisfies f (1) = 2005 and f (1)+ f (2)+ ... + f (n) = n2 f (n) for all n > 1. Find the value of f (2004). (b) If a, b are positive real numbers such that a – b = 2, then find the smallest value of the constant L for which
x
2
+ ax −
x
2
+ bx
< L for all x > 0.
(c) Let f (x) = x2 + kx ; k is a real number. The set of values of k for which the equation f (x) = 0 and f ( f (x )) = 0 have same real solution set. (d) If f (2x + 1) = 4x 2 + 14x, then find the sum of the roots of the equation f (x) = 0. Q.4
Let f (x) =
ax + b 4x + c
for real a, b and c with a ≠ 0. If the vertical asymptote of y = f (x) is x = –
vertical asymptote of y = f –1 (x) is x =
B ansal C lasses
3 4
and the
, find the value(s) that b can take on.
φ Functions & Trig.-φ φ - IV
[12]
Q.4
Classify the following functions f(x) definzed in R → R as injective, surjective, both or none . (b) f(x) = x3 − 6 x2 + 11x − 6
(a) f(x) = Q.5
Let f(x) =
(c) f(x) = (x2 + x + 5) (x2 + x − 3)
. Let f 2(x) denote f [f (x)] and f 3(x) denote f [f {f(x)}]. Find f 3n(x) where n is a natural
number. Also state the domain of this composite function. Q.6
If f(x) = sin²x+sin²
, then find (gof) (x).
Q.7
The function f(x) is defined on the interval [0,1]. Find the domain of definition of the functions. (a) f (sin x) (b) f (2x+3)
Q.8(i) Find whether the following functions are even or odd or none
(
xa
x
+ 1)
x + 1 + x 2 (a) f(x) = log
(b) f(x) =
(d) f(x) = x sin2 x − x3
(e) f(x)= sin x − cos x
(g) f(x)=
x ex − 1
x
+
2
+1
ax
(c) f(x) = sin x + cos x
−1
(f) f(x) =
(h) f(x) = [(x+1)²]1/3 + [(x −1)²]1/3
(ii) If f is an even function defined on the interval ( −5, 5), then find the 4 real values of x satisfying the
x + 1 .. x + 2
equation f (x) = f
2 + 30 π the domains (x11++2x4π)x 5 Write explicitly, functions of y defined by the following and x + find 1 x−2 x+ equations + cos x cos also and g = 1of definition x 2−x 83x + 18 3 4 of the given implicit functions :
2
Q.9
(b) x + y= 2y
(a) 10x + 10y = 10 n
− x n , x > 0 n ≥ 2 , n ∈ N , then (fof) (x) = x . Find also the inverse of f(x).
Q.10
Show if f(x) =
Q.11
(a)
Represent the function f(x) = 3x as the sum of an even & an odd function.
(b)
For what values of p ∈ z , the function f(x) = n x p , n ∈ N is even.
Q.12
a
x , 0≤ x ≤1
A function f defined for all real numbers is defined as follows for x ≥ 0 : f ( x) = [1, x > 1 How is f defined for x ≤ 0 if : (a) f is even
Q.13
(b) f is odd?
1 If f (x) = max x , for x > 0 where max (a, b) denotes the greater of the two real numbers a and b. x 1 Define the function g(x) = f(x) . f and plot its graph. x
Q.14
The function f (x) has the property that for each real number x in its domain, 1/x is also in its domain and
1 = x. Find the largest set of real numbers that can be in the domain of f (x)? x
f (x) + f
Q.15
Compute the inverse of the functions:
x + x 2 + 1 (a) f(x) = ln
B ansal C lasses
(b) f(x) = 2
x x −1
φ Functions & Trig.-φ φ - IV
(c) y =
10 x − 10 − x 10 x + 10 − x
[11]
EXERCISE–I Q.1
Find the domains of definitions of the following functions : (Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (ii) f (x) = log7 log5 log3 log2 (2x3 + 5x2 − 14x)
(i) f (x) = (iii) f (x) = ln x
− 5x − 24 − x − 2
2
(iv) f (x) =
2 log10 x + 1 − x
(vi) f (x) = log100 x
(v) y = log10 sin (x − 3) + 16 − x 2
(vii) f (x) =
1
+ ln x(x 2 − 1)
4x 2 − 1
(viii) f (x) =
log 1 2
1
(ix) f (x) = x 2 − x +
9−x
(x) f (x) =
2
(
log1 / 3 log
x x
2
−1
( x 2 − 3x − 10) . ln 2 ( x − 3) cos x −
(xi) f(x) = logx (cos 2πx )
(xiii) f(x) =
1 − 5x −x 7 −7
(xii) f (x) =
4
( [ x ]2 − 5 ) )
(xiv) f(x) =
1 2
6 + 35x − 6 x 2 1 [x]
+ log(2{x}− 5) (x 2 − 3x + 10) +
1 1− x
,
(xv) f(x) = logx sin x
1 − log 1 + 1 1 + + 10 (16 x 2 10 x) − log10 3 (xvi) f(x) = log2 log10 (log10cos2 x) − xlog 4 − log 1 / 2 ° x sec(sin 2−| x| x ) sin ( 100 ) (xvii) f (x) =
1 [x]
+ log1 – {x}(x2 – 3x + 10) +
+
(xviii) f (x) =
7 (5x − 6 − x 2 ) [{ln{x}}] + (7 x − 5 − 2x 2 ) + ln − x 2
(xix) If f(x) =
x2 − 5 x
Q.2
f + 4 & g(x) = x+ 3 , then find the domain of (x) . g
Find the domain & range of the following functions . ( Read the symbols [*] and {*} as greatest integers and fractional part functions respectively.) (i) y= log
5
(iv) f (x) =
(
)
2 (sin x − cos x) + 3
(ii) y =
x
2x
(iii) f(x) =
1+ x2
(v) y = 2 − x
1+ | x |
(vi) f (x) = log(cosec x - 1) (2 − [sinx] − [sin x]2) Q.3
−1
(vii) f (x) =
+
− 3x + 2 x2 + x − 6
x2
1+ x
x + 4 −3 x −5
Draw graphs of the following function , where [] denotes the greatest integer function. (i) f(x) = x + [x] (ii) y = (x)[x] where x = [x] + (x) & x > 0 & x ≤ 3 (iii) y = sgn [x] (iv) sgn (x −x)
B ansal C lasses
φ Functions & Trig.-φ φ - IV
[10]
13.
ODD & EVEN FUNCTIONS : If f (−x) = f (x) for all x in the domain of ‘f’ then f is said to be an even function. e.g. f (x) = cos x ; g (x) = x² + 3 . If f (−x) = −f (x) for all x in the domain of ‘f ’ then f is said to be an odd function. e.g. f (x) = sin x ; g (x) = x3 + x .
NOTE : (a) f (x) − f (−x) = 0 => f (x) is even & f (x) + f (−x) = 0 => f (x) is odd . (b) A function may neither be odd nor even . (c) Inverse of an even function is not defined . (d) Every even function is symmetric about the y −axis & every odd function is symmetric about the origin . (e) Every function can be expressed as the sum of an even & an odd function.
e.g.
(f) (g)
The only function which is defined on the entire number line & is even and odd at the same time is f(x) = 0. If f and g both are even or both are odd then the function f.g will be even but if any one of them is odd then f.g will be odd .
14.
PERIODIC FUNCTION : A function f(x) is called periodic if there exists a positive number T (T > 0) called the period of the function such that f (x+T) = f(x), for all values of x within the domain of x. e.g. The function sin x & cos x both are periodic over 2 π & tan x is periodic over π . NOTE : (a) f (T) = f (0) = f ( −T) , where ‘T’ is the period . (b) Inverse of a periodic function does not exist . f ( x ) + f ( −x ) f ( x ) − f ( − x) (c) Every constant function is always periodic, f ( x)with = no fundamental + period . 2 2 (d) If f (x) has a period T & g (x) also has a period T then it does not mean that f (x)+ g(x) must have a period T . e.g. f (x) = sinx + cosx.
15.
1
(e)
If f(x) has a period p, then
(f)
if f(x) has a period T then f(ax + b) has a period T/a (a > 0) .
f (x )
and
f (x) also has a period p .
GENERAL : If x, y are independent variables, then : (i) f(xy) = f(x) + f(y) ⇒ f(x) = k ln x or f(x) = 0 . (ii) f(xy) = f(x) . f(y) ⇒ f(x) = xn , n ∈ R (iii) f(x + y) = f(x) . f(y) ⇒ f(x) = akx . (iv) f(x + y) = f(x) + f(y) ⇒ f(x) = kx, where k is a constant .
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8.
COMPOSITE OF UNIFORMLY & NON-UNIFORMLY DEFINED FUNCTIONS : Let f : A → B & g : B → C be two functions . Then the function gof : A → C defined by (gof) (x) = g (f(x)) ∀ x ∈ A is called the composite of the two functions f & g . f (x )
→ Diagramatically → g(f(x)) . Thus the image of every x ∈ A under the function gof is the g−image of the f −image of x . Note that gof is defined only if ∀ x ∈ A, f(x) is an element of the domain of g so that we can take its g-image. Hence for the product gof of two functions f & g, the range of f must be a subset of the domain of g. PROPERTIES OF COMPOSITE FUNCTIONS : (i) The composite of functions is not commutative i.e. gof ≠ fog . (ii) The composite of functions is associative i.e. if f, g, h are three functions such that fo (goh) & (fog)oh are defined, then fo (goh) = (fog) oh . (iii) The composite of two bijections is a bijection i.e. if f & g are two bijections such that gof is defined, then gof is also a bijection. 9.
10. 11.
12.
HOMOGENEOUS FUNCTIONS : A function is said to be homogeneous with respect to any set of variables when each of its terms is of the same degree with respect to those variables . For example 5 x2 + 3 y2 − xy is homogeneous in x & y . Symbolically if , f (tx , ty) = tn . f (x, y) then f (x, y) is homogeneous function of degree n . BOUNDED FUNCTION : A function is said to be bounded if f(x) ≤ M , where M is a finite quantity . IMPLICIT & EXPLICIT FUNCTION : A function defined by an equation not solved for the dependent variable is called an IMPLICIT FUNCTION . For eg. the equation x3 + y3 = 1 defines y as an implicit function. If y has been expressed in terms of x alone then it is called an EXPLICIT FUNCTION. x →
INVERSE OF A FUNCTION : Let f : A → B be a one −one & onto function, then their exists a unique function g : B → A such that f(x) = y ⇔ g(y) = x, ∀ x ∈ A & y ∈ B . Then g is said to be inverse of f . Thus g = f −1 : B → A = {(f(x), x) (x, f(x)) ∈ f} . PROPERTIES OF INVERSE FUNCTION : (i) The inverse of a bijection is unique . (ii) If f : A → B is a bijection & g : B → A is the inverse of f, then fog = IB and gof = IA , where IA & IB are identity functions on the sets A & B respectively. Note that the graphs of f & g are the mirror images of each other in the line y = x . As shown in the figure given below a point (x ',y ' ) corresponding to y = x 2 (x >0)
changes to (y ',x ' ) corresponding to y = + x , the changed form of x = y .
(iii) (iv)
The inverse of a bijection is also a bijection . If f & g are two bijections f : A → B , g : B (gof)−1 = f −1 o g−1 .
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→C
then the inverse of gof exists and
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[8]
Into function : If f : A → B is such that there exists atleast one element in co−domain which is not the image of any element in domain, then f(x) is into . Diagramatically into function can be shown as
OR
Note that : If a function is onto, it cannot be into and vice versa . A polynomial of degree even will always be into.
Thus a function can be one of these four types : (a)
one−one onto (injective & surjective)
(b)
one−one into (injective but not surjective)
(c)
many−one onto (surjective but not injective)
(d)
many−one into (neither surjective nor injective)
Note : (i) (ii)
If f is both injective & surjective, then it is called a Bijective mapping. The bijective functions are also named as invertible, non singular or biuniform functions. If a set A contains n distinct elements then the number of different functions defined from A → A is nn & out of it n ! are one one.
Identity function : The function f : A → A defined by f(x) = x ∀ x ∈ A is called the identity of A and is denoted by I A. It is easy to observe that identity function is a bijection . Constant function : A function f : A → B is said to be a constant function if every element of A has the same f image in B . Thus f : A → B ; f(x) = c, ∀ x ∈ A , c ∈ B is a constant function. Note that the range of a constant function is a singleton and a constant function may be one-one or many-one, onto or into . 7.
ALGEBRAIC OPERATIONS ON FUNCTIONS : If f & g are real valued functions of x with domain set A, B respectively, then both f & g are defined in A ∩ B. Now we define f + g , f − g , (f .g) & (f/g) as follows : (i) (f ± g) (x) = f(x) ± g(x) (ii) (f . g) (x) = f(x) . g(x) (iii)
f f (x ) (x) = g g (x )
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domain is {x x ∈ A ∩ B s . t g(x) ≠ 0} .
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5. (i) (ii) (iii)
EQUAL OR IDENTICAL FUNCTION : Two functions f & g are said to be equal if : The domain of f = the domain of g. The range of f = the range of g and f(x) = g(x) , for every x belonging to their common domain. eg.
f(x) = 6.
& g(x) =
are identical functions .
CLASSIFICATION OF FUNCTIONS : One − One Function (Injective mapping) : A function f : A → B is said to be a one−one function or injective mapping if different elements of A have different f images in B . Thus for x1, x2 ∈ A & f(x1) , f(x2) ∈ B , f(x1) = f(x2) ⇔ x1 = x2 or x1 ≠ x2 ⇔ f(x1) ≠ f(x2) . Diagramatically an injective mapping can be shown as
OR Note : (i)
Any function which is entirely increasing or decreasing in whole domain, then f(x) is one−one . (ii) If any line parallel to x−axis cuts the graph of the function atmost at one point, then the function is one−one . Many–one function : A function f : A → B is said to be a many one function if two or more elements of A have the same f image in B . Thus f : A → B is many one if xfor ; x1, x2 ∈ A , f(x1) = f(x2) but x1 ≠ x2 . 1 2 Diagramatically a many onexmapping can be shown as
OR
Note : (i)
(ii)
Any continuous function which has atleast one local maximum or local minimum, then f(x) is many−one . In other words, if a line parallel to x−axis cuts the graph of the function atleast at two points, then f is many−one . If a function is one−one, it cannot be many−one and vice versa .
Onto function (Surjective mapping) : If the function f : A → B is such that each element in B (co−domain) is the f image of atleast one element in A, then we say that f is a function of A 'onto' B . Thus f : A → B is surjective iff ∀ b ∈ B, ∃ some a ∈ A such that f (a) = b . Diagramatically surjective mapping can be shown as
OR Note that : if range = co−domain, then f(x) is onto.
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Function (y = f (x) ) D.
ex e1/x ax , a > 0 a1/x , a > 0
(i)
logax , (a > 0 ) (a
(ii)
logxa =
1)
R+
R
R+ – { 1 }
R–{0}
R
I
R – [0, 1 )
1 , n ∈ I − {0} n
≠ 1)
[x]
(ii)
Fractional Part Functions
(i) (ii)
H.
R+ R+ – { 1 } R+ R+ – { 1 }
Integral Part Functions Functions
(i)
G.
R R–{0} R R –{0}
Logarithmic Functions
(a > 0 ) (a F.
Range (i.e. values taken by f (x) )
Exponential Functions
(i) (ii) (iii) (iv) E.
Domain (i.e. values taken by x)
{x}
R
≠1 1 [log x] a x
1 {x}
R–I
(1, ∞)
R
R+ ∪ { 0 }
R–{0}
R+
R
{–1, 0 , 1}
R
{c}
Modulus Functions
(i)
|x| 1
(ii)
I.
|x|
Signum Function |x|
,x ≠0 x =0,x=0
sgn (x) =
J.
[0, 1)
Constant Function
say f (x) = c
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4.
DOMAINS AND RANGES OF COMMON FUNCTION : Function (y = f (x) )
A.
(i)
xn , (n
N)
(ii)
, (n
∈ N)
, (n
(iv)
R = (set of real numbers)
R, if n is odd + R ∪ {0} , if n is even
R – {0}
R – {0} , if n is odd
, (n
∈ N)
∈ N)
R,
if n is odd
R+ ,
if n is even
R,
if n is odd
R+ ∪ {0} , if n is even
R+ ∪ {0} , if n is even
R – {0} , if n is odd
R – {0} , if n is odd
R+ ,
R+ ,
if n is even
if n is even
Trigonometric Functions
(i) (ii)
C.
Range (i.e. values taken by f (x) )
Algebraic Functions
(iii)
B.
Domain (i.e. values taken by x)
sin x cos x
R R
[–1, + 1] [–1, + 1]
(iii)
tan x
π∈111 / nπ π ∈ x k ∈ ,I R – (2k− +1n / ,1) n x2 2 2
R
(iv)
sec x
R – (2k + 1)
(– ∞ , – 1 ]
∪[1,∞)
(– ∞ , – 1 ] R
∪[1,∞)
(v) cosec x R – kπ , k I (vi) cot x R – kπ , k I Inverse Circular Functions (Refer after Inverse is taught ) (i)
sin–1 x
[–1, + 1]
(ii)
cos–1 x
[–1, + 1]
[ 0, π]
(iii)
tan–1 x
R
π π − , 2 2
(iv)
cosec –1x
(– ∞ , – 1 ]
(v)
sec x
(– ∞ , – 1 ]
(vi)
cot –1 x
R
–1
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∪[1,∞)
π π − 2 , 2 – { 0 }
∪[1,∞)
π [ 0, π] – 2
φ Functions & Trig.-φ φ - IV
( 0, π)
[4]
(IV)
EXPONENTIAL FUNCTION : A function f(x) = ax = ex ln a (a > 0 , a ≠ 1, x ∈ R) is called an exponential function. The inverse of the exponential function is called the logarithmic function . i.e. g(x) = loga x .
Note that f(x) & g(x) are inverse of each other & their graphs are as shown . +∞
1 a > x , a (0, 1) ) = ← x ( f
(1, 0)
(v)
f(x) = ax , 0 < a < 1
(0, 1) )45 º
x = y
+∞
)45 º
(1, 0)
x l o g a = x ) g (
x = y
→
g(x) = loga x
ABSOLUTE VALUE FUNCTION : A function y = f (x) = x is called the absolute value function or Modulus function. It is defined as
: y = x= (vi)
SIGNUM FUNCTION : A function y= f (x) = Sgn (x) is defined as follows :
y
1 for x > 0 y = f (x) = 0 for x = 0 − 1 for x < 0
y = Sgn x
y = −1 if x < 0
GREATEST INTEGER OR STEP UP FUNCTION : x if x ≥ 0 The function y = f (x) = [x] is called the greatest integer x < 0 where [x] denotes the greatest integer − x if function less than or equal to x . Note that for : −1 ≤ x < 0 ; [x] = − 1 0≤x< 1 ; [x] = 0 1≤x< 2 ; [x] = 1 2≤x < 3 ; [x] = 2 and so on . y Properties of greatest integer function : graph of y = [x] 3 (a) [x] ≤ x < [x] + 1 and 2 • º x − 1 < [x] ≤ x , 0 ≤ x − [x] < 1 1 • (b) [x+ m] = [x] +m if m is an integer . º (c) [x]+ [y] ≤ [x+ y] ≤ [x]+ [y]+ 1 • º (d) [x] + [− x] = 0 if x is an integer −3 −2 •−1 −1 1 2 x º 3 = − 1 otherwise .
•
(viii)
> x
O
It is also written as Sgn x = |x|/ x ; x ≠ 0 ; f (0) = 0 (vii)
y = 1 if x > 0
FRACTIONAL PART FUNCTION : It is defined as : g (x) = {x} = x − [x] . e.g. the fractional part of the no. 2.1 is 2.1− 2 = 0.1 and the fractional part of − 3.7 is 0.3. The period of this function is 1 and graph of this function is as shown .
B ansal C lasses
−2 −3
º
graph of y = {x}
y 1−
− −º −− • • º
• −1
φ Functions & Trig.-φ φ - IV
1
º
º −
•
•
− − − − −
x
2
[3]
KEY CONCEPTS (FUNCTIONS) THINGS TO REMEMBER : 1.
GENERAL DEFINITION : If to every value (Considered as real unless other −wise stated) of a variable x, which belongs to some collection (Set) E, there corresponds one and only one finite value of the quantity y, then y is said to be a function (Single valued) of x or a dependent variable defined on the set E ; x is the argument or independent variable . If to every value of x belonging to some set E there corresponds one or several values of the variable y, then y is called a multiple valued function of x defined on E.Conventionally the word "FUNCTION” is used only as the meaning of a single valued function, if not otherwise stated.
Pictorially :
(x ) = y f → , y is called the image of x & x is the pre-image of y under f. output
Every function from A → B satisfies the following conditions . (i) (ii) ∀ a ∈ A ⇒ (a, f(a)) ∈ f f ⊂ A x B (iii) (a, b) ∈ f & (a, c) ∈ f ⇒ b = c
and
2.
DOMAIN, CO−DOMAIN & RANGE OF A FUNCTION : Let f : A → B, then the set A is known as the domain of f & the set B is known as co-domain of f . The set of all f images of elements of A is known as the range of f . Thus : Domain of f = {a a ∈ A, (a, f(a)) ∈ f} Range of f = {f(a) a ∈ A, f(a) ∈ B} It should be noted that range is a subset of co −domain . If only the rule of function is given then the domain of the function is the set of those real numbers, where function is defined. For a continuous function, the interval from minimum to maximum value of a function gives the range.
3. (i)
IMPORTANT TYPES OF FUNCTIONS : x → input POLYNOMIAL FUNCTION : If a function f is defined by f (x) = a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an where n is a non negative integer and a0, a1, a2, ..., an are real numbers and a0 ≠ 0, then f is called a polynomial function of degree n . NOTE : (a) A polynomial of degree one with no constant term is called an odd linear function . i.e. f(x) = ax , a ≠ 0 (b)
There are two polynomial functions , satisfying the relation ; f(x).f(1/x) = f(x) + f(1/x). They are : (i) f(x) = xn + 1 (ii) f(x) = 1 − xn , where n is a positive integer . &
(ii)
ALGEBRAIC FUNCTION : y is an algebraic function of x, if it is a function that satisfies an algebraic equation of the form P0 (x) yn + P1 (x) yn−1 + ....... + P n−1 (x) y + P n (x) = 0 Where n is a positive integer and P0 (x), P1 (x) ........... are Polynomials in x. e.g. y = x is an algebraic function, since it satisfies the equation y² − x² = 0. Note that all polynomial functions are Algebraic but not the converse. A function that is not algebraic is called TRANSCEDENTAL FUNCTION .
(iii)
FRACTIONAL RATIONAL FUNCTION :
A rational function is a function of the form. y = f (x) = g (x) & h (x) are polynomials & h (x) ≠ 0.
B ansal C lasses
g(x) h (x)
, where
φ Functions & Trig.-φ φ - IV
[2]
BANSAL CLASSES TARGET IIT JEE 2008
BULLS EYE AND ACME
FUNCTIONS & INVERSE TRIGONOMETRIC FUNCTIONS Trigonometry Phase - IV
CONTENTS
FUNCTIONS KEY CONCEPT .................................................................. Page –2 EXERCISE–I ...................................................................... Page –10 EXERCISE–II ..................................................................... Page –12 EXERCISE–III ................................................................... Page –14
INVERSE TRIGONOMETRIC FUNCTIONS KEY CONCEPT .................................................................. Page –16 EXERCISE–I ...................................................................... Page –20 EXERCISE–II ..................................................................... Page –23 EXERCISE–III ................................................................... Page –25 ANSWER KEY .................................................................... Page –26 - 28
