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Apex institute for IIT-JEE is the institution of making IITians in the Ghaziabad. It is the Institute in Indirapuram to making IITians (Eng..).Its mission is to upgrade the teaching profession by providing high quality education and training to students who will be the industry's future engineers.ObjectivesThe Institute performs the five basic functions of teaching, fulfilling the following objectives:1. Highly experienced & highly qualified recruitments and specialized training of faculty members.2. Providing study material (specially in house designed) as per requirement3. Teaching methodology and conduct periodical exams.4. Framing of test papers based on Latest Examination pattern.5. Course Co-ordination.

### Transcript of Function ASSIGNMENT FOR IIT-JEE

1. The period of the function, f(x) = [sin 3x] + |cos 6x| is : ( [.] denotes the greatest integer less than or equal to x)(a) (b) 32(c) 2 (d) 22. The function f(x) = [ ]sec1xx x , where [x] denotes the greatest integer less than or equal to x is defined for allx belonging to :(a) R (b) R - {(-1, 1) {n : n I)} (c) R - (0, 1) (d) R - {n : n N}3. The function f(x) = ) x ( log 2x is defined for x belonging to :(a) (- , 0) (b) (1,) (c) (0, ) (d) none of these4. If f(x) = , 145g and3x cos . x cos3x sin x sin 2 2 ,_

,_

+ + ,_

+ + then (gof)(x) =(a) 1 (b) -1 (c) x (d) none of these5. Let [x] denote the greatest integer x . The domain of definition of function 2 ] x [x 4) x ( f2+ is(a) ] 2 , 1 [ ) 2 , ( (b) [0, 2] (c) [-1, 2] (d) (0, 2)6. The domain of definition of the function

,_

4x x 5log ) x ( f210 is(a) [1, 4] (b) (1, 4) (c) (0, 5) (d) [0, 5]7. The range of the function x 3 cos 21) x ( fis(a) 1]1

0 ,31(b) R (c) 1]1

1 ,31(d) none of these8. The domain of definition of the function x | x |1) x ( f is(a) R (b) (0, ) (c) (- ,0) (d) none of theseLEVEL - 1 (Objective)FUNCTION9. The function ) 1 x x log( ) x ( f 2+ + is(a) an even function (b) an odd function (c) periodic function (d) none of these10. The function ( ) ) 1 x x log( cos ) x ( f 2+ + is(a) even (b) odd (c) constant (d) none of these11. Thr period of the function f(x) = sin4x+ cos4x is(a) (b) /2 (c) 2 (d) none of these12. Which of the following functions is an odd function(a) f(x) = constant (b) f(x) = sinx + cosx (c) ( ) ) 1 x x log( sin ) x ( f 2+ + (d) f(x) = 1 + x + 2x313. The domain of definition of the functionf(x) = 7-xPx-3 is(a) [3, 7] (b) {3,4,5,6,7} (c) {3,4,5} (d) none of these14. The function x tan x xx cos x sin) x ( f24 4++ is(a) even (b) odd (c) periodic with period (d) periodic with period 215. Range of f(x) = |x| + |x+1| is(a) [0, ) (b) (0, ) (c) (1, ) (d) [1, )16. Range of tan-1x - cot-1x is(a) (0, ) (b) 1]1

2,23(c) ,_

1 ,32(d) 1]1

1 ,3217. The range of x | x | is(a) (0,) (b) [0,) (c) (-,0) (d) (-,0]18. If

,_

+1 101 10x ) x ( fx 2x 22 then f is(a) an even function (b) an odd function (c) neither even nor odd (d) cannot be determined19. The domain of the function f(x) = ) x 1 ( log110 + ) 2 x ( + is(a) [-3, -2] excluding (-2.5) (b) [0, 1] excluding 0.5(c) [-2, 1], excluding 0 (d) None of these20. The period of x cos ] x [ x x cos 2 4e + + is ([.] denotes the greatest integer function) (a) 2 (b) 1 (c) 0 (d) -121. The domain of the function f(x) =

,_

++ + x 2x 1sin ) x cos(sin ) x (log sin2121(a) {x : 1 < x < 2} (b) {1} (c) Not defined for any value x (d) {-1, 1}22. If [.] denotes the greatest integer function then the domain of the real valued function is | 2 x x | log 2] 2 / 1 x [ +(a)

,_

,23(b) ) , 2 ( 2 ,23

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(c) ) , 2 ( 2 ,21

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(d) None of these23. The domain of the function f(x) = 6) 1 x ( 2) 2 x (32x2 52 8 4 + is(a) (0, 1) (b) [3, ) (c) (1, 0) (d) none24. A function whose graph is symmetrical about the y-axis is given by(a) ) 1 x x ( log ) x ( f 2e + + (b) f(x + y) = f(x) + f(y) for all x, y R(c) f(x) = cos x + sin x (d) none of these25. If

,_

+ ,21x , 1 x x ) x ( f 2 then value of x satisfying f(x) = f -1(x) is(a) 1 (b) 2 (c) 21(d) none of these26. If ,_

5 x1 xlog ) x ( f 4 . 0 and g(x) = x2 - 36, then Df/g is(a) } 6 { ~ ) 0 , ( (b) } 6 , 1 { ~ ) , 0 ( (c) } 6 { ~ ) , 1 ( (d) } 6 { ~ ) , 1 [ 27. The domain of the real-valued function f(x) = loge |loge x| is(a) ) , 1 ( ) 1 , 0 ( (b) ) , 0 ( (c) ) , e ( (d) ) , 1 ( 28. If f(x) and g(x) are two functions of x such that f(x) + g(x) = ex and f(x) - g(x) = e-x, then(a) f(x) is odd, g(x) is odd (b) f(x) is even, g(x) is even(c) f(x) is even, g(x) is odd (d) f(x) is odd, g(x) is even29. The inverse of the function ) 1 a , 0 a ( ) 1 x x ( log y 2a > + + is(a) ) a a (21 x x (b) not defined for all x(c) defined for only positive x (d) none of these30. Let f(x) = cos p x, where p = [a] = the greatest integer less than or equal to a. If the period of f(x) is , then(a) ] 5 , 4 [ a (b) ] 5 , 4 [ a (c) ) 5 , 4 [ a (d) none of these31. If f(x) = , e | x n |cos ........ | x 2 |cos | x |cos ] x [ x + + + + then period of f(x) is(a) 1 (b) n1(c) 1 n1 n22+(d) n ...... 3 . 2 . 1132. Let

,_

+ ,_

+ ,_

4| x | 2tan4| x | 2cos4| x | 2sin ) x ( f 1 1 1, then Df is(a) [-6, 6] (b) ) , 6 [ (c) [-6, 3] (d) [-3, 6]33. Identify the statement(s) which is/are incorrect ?(a) The function f(x) = cos(cos-1 x) is neither odd nor even(b) The fundamental period of f(x) = cos(sin x) + cos(cos x) is (c) The range of the function f(x) = cos (3 sin x) is [- 1, 1](d) None of these34. Let 9x cos 4 ) x ( f22 . Then(a) ] 1 , 1 [ R , , ,3D f f ,_

(b) ] 2 , 2 [ R , , ,3D f f ,_

(c) ] 4 , 4 [ R and ,3 3, D f f ,_

1]1

(d) ] 4 , 0 ( R ,3, D f f ,_

35. The range of the function ) , 1 ( x ), x x xe sin( ) x ( f 2 ] x [ + where [x] denotes the greatest integer function is:(a) (b) [0, 1] (c) [-1, 1] (d) R36. The domain of f(x) = log2 log3 / 4log (tan-1 x)-1 =(a) R (b) ) , / 4 ( (c) (0, 1) (d) None of these37. If f(x) is a periodic function of the period then f(x + a), where a is a constant, is a periodic function of theperiod(a) 1 (b) (c) a(d) none of these38. If ] 1 x [1| x |11 ) x x ( cos ) x ( f22 1+

,_

+ then domain of f(x) is (where [.] is the greatest integer)(a) 1]1

+25 1, 2 (b) 1]1

+25 1, 2 (c)

,_

22 1, 2 (d) none of these39. Let f(x) = sin x, g(x) = ln |x|. If the ranges of the composite functions fog and gof are R1 and R2 respectively, then(a) ) 0 , ( R ), 1 , 1 ( R 2 1 (b) ] 1 , 1 [ R ], 0 , ( R 2 1 (c) ] 0 , ( R ), 1 , 1 ( R 2 1 (d) ] 0 , ( R ], 1 , 1 [ R 2 1 40. If x 1x 1log ) x ( f+ , then(a) f(x) is even (b) f(x1).f(x2) = f(x1 + x2) (c) ) x x ( f) x ( f) x ( f2 121 (d) f(x) is odd1. Find the value of x for which, 0) 4 x () 2 x ( ) 1 x )( 1 x 2 () x ( f43 2> 2. Find the values of x for which 0) 1 x () 4 x ( ) 3 x ( ) x 1 ( ) 2 x () x ( f2 2+ .3. Solve 12 xx | 3 x |>++ +.4. Find the domain of the function;

,_

+ | x sin | log3) 23 x 8 x ( log log ) x ( f22| x sin |5. Find domain for

,_

+5 x1 xlog ) x ( f 4 . 0.6. Find domain for x log3| x | 2 1cos y | 1 x |1+

,_

.7. Find domain for

,_

] 1 x [cos ] x [sin ) x ( f where [ ] denotes greatest integer function.8. Find the range for x ] x [ 1] x [ xy+ where [ ] denotes greatest integer function.9. Find domain and range of the function y = loge(3x2 - 4x + 5).10. If f is an even function, find the real values of x satisfying the equation ,_

++2 x1 xf ) x ( f .11. Find whether the given function is even or odd function, where 21 x) x tan x (sin x) x ( f1]1

++ .where [ ] denotesgreatest integer function.LEVEL - 2 (Subjective)312. If ) , 2 [ ) , 1 [ : f is given by x1x ) x ( f + then find ) x ( f 1 . (assume bijective).13. Let ) , 4 / 3 [ ) , 2 / 1 [ : f , where f(x) = x2 - x + 1. Find the inverse of f(x).14. ,x ] x [1) x ( f where [ ] denotes greatest integral function less than or equals to x. Then find domain of f(x).15. Find the domain of ) 13 x 7 x ( log1) x ( f22 / 1 + 16. Find the domain of single valued function y = f(x) given by the equation 10x + 10y = 10.17. Let ,_

2, 0 x, then find the solution of the function x tan log1) x ( fx sin .18. Find the range of log3(log1/2(x2 + 4x + 4)).19. Find the domain & range of : 22x9sin 3 ) x ( f .20. Find the inverse of following functions:(i) ] 3 , 3 [ x ), 3 / x ( sin ) x ( f 1 [assuming bijective](ii) ] 3 , 1 [ x ), 1 x 3 x ( ln ) x ( f 2 + + . [assuming bijective]21.' 1 x 0 , x0 x 1 , 1 x) x ( f2 and g(x) = sinx. Find h(x) = f(|g(x)|) + |f(g(x))|.22. If , x ] cos[ x ] cos[ ) x ( f 2 2 + where [x] stands for the greatest integer function, then evaluate) 4 / ( f and ) ( f ), ( f ), 2 / ( f .23. A cubic expression f(x) satisfies the condition

,_

,_

+x1f ) x ( fx1f ) x ( f , then prove that f(x) = 1 + x3or1 - x3.If f(3) = 28. Then prove that f(2) = 9.24. Let f(x) be a polynomial function satisfying, R y , x 2 ) xy ( f ) y ( f ) x ( f ) y ( f ) x ( f + + . If f(2) = 5 thenprove that f(5) = 26.25. If for non-zero x, 5x1x1bf ) x ( af

,_

+ where b a then find f(x).1. The domain of the function 1 x ) x 1 log( ) x ( f 2 + is(a) [-1, 1] (b) ) , 1 ( (c) (0, 1) (d) ] 1 , ( 2. The range of the function 22xx 1) x ( f + is equal to(a) [0, 1] (b) (0, 1) (c) ) , 1 ( (d) ) , 1 [ 3. The curves 2 x 3 x y and 2 | x | 3 | x | y 2 3 2 3+ + + + have the same graph for(a) x > 0 (b) 0 x (c) all x except 0 (d) all x4. Domain of the function 73 x12x1) x ( f x sin21++ + is(a) (b) R - {0} (c) R (d) None of these5. The domain of definition of the function ) 1 x ( log e 3 y 1 x2 is(a) ) , 1 ( (b) ) , 1 [ (c) R ~ {1} (d) ) , 1 ( ) 1 , ( 6. The range of the function f(x) = cos [x], where 2x2< 0, then domain of the function y = log (ax3 + (a + b)x2 + (b + c)x + c) is(a) ;'a 2b~ R2(b)

,_

;' } 1 x | x {a 2b~ R(c)

,_

;' ] 1 , (a 2b~ R (d) none of these8. Which of the following functions is an even function?(a) x xx xa aa a) x ( f+ (b) 1 a1 a) x ( fxx+ (c) 1 a1 ax ) x