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FUNCTIONS
[ DOMAIN, RANGE, PERIOD, INVERSE, EVEN or ODD ]
By:- Nishant Gupta
For any help contact:
9953168795, 9268789880
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Funções: Tipos, Domínio, Código de Exemplos e Aplicações, Exercícios de Matemática
Este documento aborda os conceitos básicos de funções matemáticas, incluindo seus tipos, domínio, intervalo e alguns exemplos. Além disso, discutimos aplicativas práticas de funções em diferentes contextos.
Tipologia: Exercícios
2013
1 / 9
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FUNCTIONS
[ DOMAIN, RANGE, PERIOD, INVERSE, EVEN or ODD ]
By:- Nishant Gupta
For any help contact:
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
1. Definition: Let A & B be two sets. A relation from A to B is called a function if for each a A there exists
one & only one b B such that ordered pair (a, b) f. Denoted f : A B
A is called Domain B is called Co domain
b is called image of a under f Range : set of images of f
2. Different types of functions
1-1 or injective : If for x 1 , x 2 A ; x 1 x 2 f(x 1 ) f(x 2 )
Onto or surjective : If for all b B there exists a A such that f(a) = b
Into : If there is at least one b B for which there is no a A such that f(a) = b
Bijective : If a function is both 1-1 & onto then it is known as bijective
IDENTITY FUNCTION : If f (x) = x for x R
EVEN & ODD FUNCTIONS f (-x) =
f(x) ODD
f(x) EVEN
3. If f : A B A & B contain n & m elements resp then
Number of 1 – 1 functions from A to B are
0 if m n
P if m n n
m
Number of onto functions are
m
r 1
n r
m r m 1 C r
NOTE : If B contains 2 elements then number of onto functions is 2 n^ – 2
Total number of functions is m n
4. Step function or GREATEST INTEGER FUNCTION [x] :
It means greatest integer x i.e. [2.3] = 2 & [-2.3] = - 3
5. {x} ,Fractional part of x {1.4} = 0.4 {-1.4} = 0.
6. Signum Function :
0 ifx 0
1 ifx 0
1 ifx 0
x
|x|
7. Lograthims:
'log' is involved in many ways so let's be thorough about it
log x is defined for x > 0 only
logA B = x means B = A x
FUNCTIONS
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
In General fog (x) gof (x) ; but fo(goh) = (fog)oh and (gof)
= f - og -- n (A) = 4 & n(B) = 6 then no. of injections
from A to B is
(a) 360 (b) 6^4
(c) 4^6 (d) 120
- If n(A) = 5 & n (B) = 2 then no. of surjections
from A to B is
(a) 23 (b) 30
(c) 12 (d) N/T
- No. of bijective function. From A to itself when
a contains 106 elements is
(a) 106! (b) (106)^2
(c) 2 106 (d) 2 106
- 1
- Domain of cos x 2 is
(a) R (b) [0, )
(c) {-/2, /2] (d) N/T
- Domain of x 1 log ( 1 x)
10
is
(a) [-3, -2] – {- 2.5} (b) [0,1] – {.5}
(c) [-2, 1) – {0} (d) N/T
- Domain of log x
- 5
(a) (0,1] (b) (0, )
(c) (.5, ) (d) N/T
- Domain of Px 2 6 x
(a) {2,4} (b) {1,4}
(c) {2, 6} (d) None.
Domain of sin
x + sec - x
(a) [ - 1, 1] (b) R – [-1, 1]
(c) {- 1, 1} (d) None.
- Domain of log x x , {. } is fractional part, is
(a) N – {1} (b) R – {1}
(c) R - I (d) N/T
- Domain of
1 / 2 2
3 4
x 3 x log
is
(a) ( -1 , 4 )
(b) ( 1 , 4 )
(c) ( -∞ , -1 ) ( 4 ,∞ )
(d) ( -∞ , 0 ) ( 3 ,∞ )
- Domain of f(x) = 3 x 7
183 x 2 x 3
15 x C P
is
(a) {2, 3} (b) {3,4,}
(c) {2, 3 ,4} (d) {2, 3,4,5}
- Domain of log( 2 x x )
2 is
(a) (0, ) (b) (1, )
(c) (0,1) ( 1 ,) (d) N/T
- Domain of f(x) = log 2 log 3 log 4 x is
(a) 2 < x < 4 (b) x < 4
(c) x > 4 (d) 5 >x > 4.
- Domain of log ( log
x
x ) , {x} denotes
fractional part , is
(a) R (b) R - I
(c) ( 1 , ∞ ) - I (d) ( -∞ , 1) – I
- Domain of (^2 x)x [x]
is
(a) [ 0 , 2] (b) [0,1]
(c) [ 1, 2] (d) N/T
Domain of Sin
x log 3 is
(a) [-1,9] (b) [1,9]
ASSIGNMENT
FUNCTIONS
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
(c) [-9,1] (d) N/T
- If f(x) = x then domain of fof (x) is
(a) (0, ) (b) (-, 0)
(c) {0} (d) N/Ts
- Domain of Cos-1
2 |x| (^) + (log ( - x + 3))-1 (^) is
(a) (-2, 6) (b) [-6, 2] [2,3]
(c) [-6, 2] (d) N/T
- Domain of log|log x| is
(a) (0, ) (b) (1, )
(c) (0,1) ( 1 ,) (d) N/T
- Domain of f(x) = 16 x x log x 1
2
2
1
2 is:
(a) [0, 16] (b) (1, )
(c) (1, 16) (d) [1, 16]
- Domain of f(x) = 3 x 7
183 x 2 x 3
15 x C P
is
(a) {2, 3} (b) {3,4,}
(c) {2, 3 ,4} (d) {2, 3,4,5}
- Range of cos- 1 x
1 x cosec 1 x
1 x 1
(^) is
(a) [- / 2 /2] (b) { / 2}
(c) {0, / 2} (d) N/T
- Range of [Sin x] is
(a) [-1, 1] (b) {-1, 1}
(c) (-1, 1) (d) N/T
- Range of [Sin x] , x (-/4, /4) is
(a) [-1, 1] (b) {-1, 1}
(c) (-1, 1) (d) N/T
- Range of cos([x] (^) )
(a) 0 (b) {0}
(c) {-1,1} (d) N/T
- Range of [Sin x] , x (-/4, /4) is
(a) [-1, 1] (b) {-1, 1}
(c) (-1, 1) (d) N/T
- Range of
2
e e
x x is
(a) R (b) [0, )
(c) [1, ) (d) None.
- Range of , x 0 1 [x]
e
x
(a) ( 0 ,∞ ) (b) [ 0 , ∞ )
(c) ( 1, ∞ ) (d) [1 , ∞ )
- Range of f ( x ) = 5 | sinx | - 3|cos x | is
(a) [ 3, -5 ] (b) [ - √34 , √34 ]
(c) [ 5 , √34 ] (d) [ -3 , 5 ]
- f : R R , f (x ) = 2
e e
x x is
(a) many one onto (b)neither 1-1 nor onto
(c) one-one onto (d) one-one into
- If f ( x) = acos ( bx + c) + d then range is
(a) [ d+a , d+2a ] (b) [ a-d , d+a ]
(c) [ d+a , a-d ] (d) [d-a , d+a ]
- Range of 1 |x|
x
(a) [-1, 1] (b) R
(c) (-1, 1) (d) N/T
- Range of x 5
9 x P (^)
is
(a) {5,7} (b) {1,2}
(c) {3,4,…7} (d) N/T
- Range of 2 cos 3 x
is
(a) [1/3,1] (b) [1/2,2]
(c) [1/2,1] (d) N/T
- Period of tan 4
x cos 3
x is
(a) 12 (b) 16
(c) 24 (d) N/T
- Period of sin
2 x 3 is
(a) 2 (b) 6
(c) 6 2 (d) None
- Period of sin 2 x + sin 4 x/2 + sin 8 x/4 is
(a) π (b) 4π
(c) 2π (d) 8π.
- If f is a periodic functions , then
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
(c) 5 - (1+sin^2 x)^3 (d) N/T
- sin
- 1 (sin12) + cos - (cos12) is equal to
(a) 0 (b) 24-2
(c) 4 -24 (d) N/T
- Which of the followings are odd
(a)
e cosx
x
x
(b) x^2 ln ( x 1 x )
6 3
(c) g x g x
x
x
(d)g x g x h x h x
- If the real -valued function
1
n x
x
x a
a fx f is even, then n equals
(a) 2 (b) 2/
(c) 1/4 (d) -1/
upto infinite terms is ( GIF )
(a) 2006 (b) 2007
(c) 2008 (d) 2009
- If f ( x ) is polynomial satisfying f ( x ) =
1 f( 1 /x)
f(x) f( 1 /x) f(x)
& f ( 2 ) = 17 then
1
1
f(x) dx is equal to
(a) 0 (b) 5/
(c) 12/5 (d) 6/
- The fundamental period of f ( x ) = ( sin x )^0 +
( cos x )^0 + ( tan x )^0 is
(a) 2π (b) π
(c) π /2 (d) Does not exist
- f : N N where f ( x ) = x + (-1) x^ , then f is
(a) 1-1 into (b) Many one into
(c) 1-1 onto (d) N/T
- If f ( x ) = 2
x
e 1
xe
x
x
is
(a) Even (b) Odd
(c) Neither (d) Both
- If f ( x ) =
sin x
0
2 tdt then period of f / ( x ) is
(a) π/2 (b) π
(c) 2π (d) 3π/
- Period of f ( x ) = [ 8x + 7 ] + | tan 2π x + cot
2πx | - 8x is
(a) 1/2 (b) 1
(c) 1/4 (d) N/T
66. A B
is a
(a) 1 – 1, on to (b) 1 – 1, into
(c) Many – 1, onto (d) N/T
- If log 2 log 3 log 4 x = 1 then x is
(a) 4 9 (b) 3 4
(c) 2^9 (d) N/T
, GIF
(a) 24 (b) 25
(c) 26 (d) N/T
(a) 498 (b) 499
(c) 500 (d) 501
- Let f( ) = Sin (Sin +Sin3 ) then f( ) is
(a) 0 if 0 (b) 0 for all
(c) 0 for all (d) N/T
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-
- If f (x+2y, x-2y) = x y then f(x , y) is
(a) 8
x y
2 2 (b) 4
x y
2 2
(c) 8
x y
2 2 (d) N/T
- If f (x) = (a -.x n)1/ n, then f [f (x)] is equal to
(a) xn^ (b) a – x
(c) x (d) n x.
- f : R R is f (x) = 2x + sinx, x R
(a) 1 – 1 & on to (b) 1 – 1, into
(c) 1 + 1 into (d) neither 1 1nor into
- If f(x) = cos (log x ) then f(x)f(y) -
f(xy) y
x f 2
is
(a) 1 (b) ½
(c) – 2 (d) N/T
- f(x) = cos[ 2 ]x + cos[- 2 ]x , GIF then
(a) f(/2) = 0 (b) f( ) = -
(c) f(- ) = 0 (d) N/T
- If f : [0,1] B where f (x) = x 2
- 1, is
surjective then B is
(a) [1,4] (b) [1,2]
(c) [1,∞) (d) N/T
- If f (x+1) - f (x-2) = 0 , period f (x) is
(a) 2 (b) 3
(c) 1 (d) Does not exist
- Domain of
2
1
sin sin x
x x is
(a) (0, ) (b) (-∞, )
(c) [ 2nπ , (2n+1) π ], n I (d) N/T
- Period of 2 | sin2x | - 3 | cos3x | is
(a) π (b) π/
(c) π/2 (d) N/T
- Range of y =
x x
e
x
is
(a) [1,e] (b) [1,∞)
(c) ( 0, ∞) (d) N/T
- f : R R , defined by f( x) = e x is
(a) 1-1 , onto (b) many 1, into
(c) 1-1, into (d) many 1, onto
- |sin x| has inverse if its domain is
(a) [0, π] (b) [0, π/2]
(c) [- π/4, π/4] (d)N/T
- If f: RR, g :R R are functions defined by f
(x) = x + l, g (x) = x^2 - 1 then g { f (3)}will be
(a) 9 (b)
(c) 15 (d) 18
- Let f (x ) = x 1
x
, x 1 then for what value
of α , fof ( x ) is identity function
(a) √2 (b) - √
(c) 1 (d) -
- If f ( x ) =
n (^) n a x , x > 0 then on its
domain f( x) is
(a) only injective (b) only surjective
(c) bijective (d) N/T
- If f ( x ) = sinx + cosx , g ( x ) = x 2
- 1 , then g
( f ( x) ) is invertible in domain
(a) [0, π/2 ] (b) [- π/2 , π /2]
(c) [ - π/4 , π /4 ] (d) [ 0, π ]
- Number of maps f from set {1, 2,3} into {1, 2,
3, 4, 5} such that f (i) ≤ f (j) whenever i <j is
(a) 60 (b) 50
(c) 30 (d) 35.