FUNCTION Sets and their Representations, Union, intersection and complements of sets, and their algebraic properties, Relations, equivalence relations, mapping, one-one, into and onto mappings, composition of mappings. Polynomials, rational, trigonometric, logarithmic and exponential functions, inverse functions. Graph of simple functions. INITIAL STEP EXERCISE 1. Let E = {1, 2, 3, 4} and F = {1, –2} then the number of onto functions from E to F is (a) 14 (b) 16 (c) 12 (d) 8 The domain of the function (a) R – {–1, –2} (b) (c) (d) 3. (a) (b) (c) (d) 4. (–2, ∞) R – {–1, –2, –3) (–3, ∞) – {–1, –2} ≥ 0 only when θ ≥ 0 ≤ 0 for all real θ ≥ 0 for all real θ ≤ 0 only when θ ≥ 0 (–∞, ∞) (– ∞, 3 – √3) ∪ (3 + √3, ∞) ( – ∞, 1] ∪ [5, ∞)
f (x) = log 2 ( x + 3) x 2 − 3x + 2 .
2.
Let f(θ) = sin θ (sin θ + sin 3θ) then f(θ)
2 The domain of the function log( x − 6x + 6) is
(a) (b) (c) (d) 5. (a) (c) 6.
(0, ∞) If f(x) = x2 – x–2 then f(1/x) is equal to f(x)
1 f (x)
(b) (d)
–f(x) [f(x)]2
If f(x) = (a – xn)1/n then f(f(x)) equals. (a) x (b) a–x
(c) 7.
xn
(d)
x1/n
If f(1) = 1 and f(x + 1) = 2 f(x) + 1 if x ≤ 1 then f(x) is (a) 2x + 1 (b) 2x (c) 2x – 1 (d) 2x – 1 – 1
⎛ π π⎞ ⎜− , ⎟ If 1 + 2x is a function having ⎝ 2 2 ⎠ as domain and (–∞, ∞) as codomain then it is
8.
(a) (b) (c) (d) 9.
onto but not one-one one-one but not onto one-one and on to neither one-one not on to
f (x) = a x −1 x n (a x + 1) is
If the real valued function (a) 2 (b) (c) 1/4 (d)
even then n equals
2/3 –1/3
f (x) = 2x − 2− x 2 n + 2 − x is
10.
11.
Range of the function then (a) (–1, 1) (b) [–1, 1] (c) (0, 1] (d) none of these If [x] stands for the greatest integer function, then the value of
1 ⎤ ⎡1 2 ⎤ ⎡1 ⎡ 1 999 ⎤ ⎢ 2 + 1000 ⎥ + ⎢ 2 + 1000 ⎥ + .... ⎢ 2 + 1000 ⎥ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣
(a) (c)
f (x) =
498 500
x
(b) (d)
499 501
12.
If (a)
1 + x 2 then (fofof) (x) = 3x 1 + x (b)
2
x 1 + 3x
2
3x
(c)
1 − x 2 (d)
none of these
13.
14.
If f(x) = 9x2 + bx + c then the value of a and b for which the identify f(x + 1) – f(x) = 8x + 3 is satisfied are (a) a = 1, b = 4 (b) a = 1, b = –4 (c) a = 9, b = 1 (d) a = 4, b = –1 Let P(x) = a2 + bx, g(x) = lx2 + mx + n, if P(1) – g(1) = 0, P(2) – g(2) = 1, P(3) – g(3) = 4 then P(4) – g(4) equals. (a) 0 (b) 5 (c) 6 (d) 9
f ( x ) = sec−1
15.
The function (a) all real x (b) (c) (d)
x x − [x]
is defined for
R – {(–1, 1) ∪ Z} R+ – (0, 1)
16.
R+ – Z If f(x) = x – x2 + x3 – x4...........to ∞ for |x| 0 ⎩
1.
Let g(x) = 1 + x – [x] and (a) x (b) (c) f(x) (d)
then for all x, f [g(x)] is equal to
1 g(x)
2.
3.
Let f : R → R be any function. Define g : R → R by g(x) = |f(x)| for all x then g is (a) onto if f is onto (b) one-one if f is one. (c) continuous if f is continuous (d) differentiable if f is differentiable. The domain of the function y(x) given by the equation 2x + 2y = 2 is (a) (c) 0
4.
5.
The function f : R → R defined by f(x) = (x – 1) (x – 2) (x – 3) is (a) one-one but not onto (b) onto but not one-one (c) both one-one and on to (d) neither one-one nor onto If the function f : [1, ∞) → [1, ∞) is defined by f(x) = 2x (x – 1) then f–1(x) is (a) (b) (c) (d)
⎛1⎞ ⎜ ⎟ ⎝2⎠
x ( x −1)
1 1 + 4 log 2 x 2
[
]
}
1 1 − 1 + 4 log 2 x 2
{
not defined
6.
If g(f(x)) = |sin x| and f(g(x)) = (sin √x)2 then (a) f(x) = sin2x, g(x) = √x (b) (c) (d) f(x) = sin x g(x) = |x| f(x) = x2, g(x) = sin √x f and g cannot be defined
7.
If f(x) = 1 + αx, α ≠ 0 is the inverse of itself then the value of α is (a) –2 (b) –1 (c) 0 (d) 2
8.
⎛ | sin x | sin x ⎞ f ( x ) = ±⎜ ⎜ cos x + | cos x | ⎟ ⎟ ⎝ ⎠ is periodic with period The function
(a) 9.
π
(b)
2π
(c) π/2 (d) none of these If f(x) is an odd periodic function with period 2 then f(4) equals (a) 0 (b) 2 (c) 4 (d) –4
⎛π ⎞ ⎛π ⎞ ⎜ + x⎟ ⎜ + x⎟ 2x + cos2 ⎝ 3 ⎠ – cos x . cos ⎝ 3 ⎠ constant (independent If the function f(x) = cos
10.
of x) then the value of this constant is (a) 0 (b) 3/4 (c) 1 (d) 4/3 11. The range of the function sin(π[x]) is (a) 0 (b) {0} (c) [–1, 1] (d) [0, 1] f(x) = ex – [x] + |cos πx| + |cos 2πx| + .....|cos nπx] then period of f(x) is (a) (c) 13. 1
n −1 n2 +1
2
12.
(b) (d)
1 n 1 1.2.3...n
⎛ 2− | x | ⎞ −1 cos −1 ⎜ ⎟ + [log(3 − x ] ⎝ 4 ⎠ The domain of the function f(x) = is
(a) (c) 14.
[–6, 6] (2, 3)
(b) (d)
(–∞, 2) ∪ (2, 3) [–6, –2) ∪ (2, 3)
f ( x) f
If f(x) is a polynomial satisfying (a) 126 (b) –125 (c) 26 (d) none of these
FG 1 IJ = f ( x) + f FG 1 IJ H xK H xK
and f(3) = 10, then f(5) =
15.
Let f(x) = cos √px, where p = [a] = the greatest integer less than or equal to a. If the period of f(x) is π then (a) a ∈ [4, 5] (b) a = 4, 5 (c) a ∈ [4, 5) (d) none of these Let f(x) be a polynomial function of the second degree. If f(1) = f(–1) and a1, a2, a3 are in AP then f’(a1), f’(a2), f’(a3) are in (a) (c) AP HP (b) (d) GP none of these
16.
17.
1 ⎡ ⎛ x 2 ⎞ ⎛ x 2 ⎞⎤ f ( x 2 )f ( y 2 ) − ⎢f ⎜ ⎟ + f ⎜ 2 ⎟ ⎥ 2 ⎣ ⎜ 2 ⎟ ⎜ y ⎟⎦ ⎝ ⎠ ⎝ ⎠ has the value If f(x) = cos (log x), then
(a) (c) 18. (a) 19.
–2 ½ π
(b) (d) (b)
–1 none of these 2π
Fundamental period of the function f(x) = sin4x + cos4x = (c) π/2 (d) None If f(x) is an odd periodic function with period 2, then f(4) equals (a) 0 (b) 2 (c) 4 (d) –4 If 2f(x) – 3f(1/x) = x2, x is not equal to zero, then f(2) is equal to (a) (c) 5/2 –1 (b) (d) –7/4 none of these
20.
21.
The graph of the function cos x cos (x + 2) – cos2 (x + 1) is (a) a straight line passing through (0, –sin21) with slope 2 (b) (c) (d) axis. a straight line passing through (0, 0) a parabola with vertex (1, –sin21) a straight line passing through the point
⎛π 2 ⎞ ⎜ ,− sin 1⎟ ⎝2 ⎠ are parallel to the x-
22.
Domain of the function f(x) = arc sin [log2 (x2/2)] is (a) (c) [–2, 2] (b) (–1, 1) (d) [0, 2] [–2, –1] ∪ [1, 2]
23.
2 ⎡ ⎛x⎞ ⎤ sin −1 ⎢log 2 ⎜ ⎟ ⎥ ⎝2⎠ ⎥ ⎢ ⎣ ⎦ is given by Domain of
(a) (c) 24.
[–2, –1] [–2, –1] ∪ [1, 2]
(b) (d)
[1, 2] none of these
⎛1+ x2 ⎞ f ( x ) = cos(sinx ) + sin −1 ⎜ ⎟ ⎜ 2x ⎟ ⎠ is defined for ⎝ The function
(a) (c) 25.
x = –1, 1 x∈R
(b) (d)
x ∈ [–1, –1] x ∈ (–1, 1)
f (x) = 3 + log 10 ( x 3 − x ) 4 − x2 is
Domain of definition of the function (a) (b) (–1, 0) ∪ (1, 2) (1, 0) ∪ (2, ∞)
(c) (d) 26.
(–1, 0) ∪ (1, 2) ∪ (2, ∞) (1, 2)
If f : R → R satisfies f(x + y) = f(x) + f(y), for all x, y ∈ R and f(1) = 7 then
7(n + 1) 2 (b) 7 n (n + 1) 2 7n 2
∑ f (r)
r =1
n
is
27.
(a) 7n(n + 1) (c) (d) The real number x when added to its inverse gives the minimum value of the sum of x equal to (a) 1 (b) –1 (c) –2 (d) 2 The inverse of the function (a) (b) (c) (d)
log 10 ( 2 − x )
28.
f (x) =
10 x − 10 − x 10 x + 10 − x is
1 ⎛1+ x ⎞ log 10 ⎜ ⎟ 2 ⎝1− x ⎠
1 log10 (2 x − 1) 2
1 ⎛ 2x ⎞ log ⎜ ⎟ 4 ⎝2−x ⎠
29.
⎛ 2 log x + 1 ⎞ f ( x ) = log 100 x ⎜ ⎟ ⎝ − x ⎠ is The domain of definition of
(a) (b) (c) (d) 30.
(0, 10–2) ∪ (10–2, 10–½) (0, 10–½) (0, 10–1) none of these
31.
The domain of definition of the function f(x) = log4 log5 log3 (18x – x2 – 77) is (a) [8, 10] (b) (8, 10) (c) [1, 10] (d) none of these If [x] and {x} represent integral and fractional parts of x, then the expression
[x] + ∑
2000
{x + r} r =1 2000 is equal to
32.
(a) x + 2001 (c) x The period of the function f(x) = |sin x| + |cos x| is (a) (c) π/2 2π (b) (d) π none of these
2001 x 2 (b)
(d)
[x] +
2001 2
f (x) =
33.
The domain set of definition of (a) (b) (c) (d)
(−∞, ∞) − [−2, 2] (−∞, ∞) − [−1,1] [−1,1] ∪ (−∞, 2) ∪ (2, ∞)
1− | x | 2− | x | is
none of these
34.
The domain of the function f ( x ) = log10 ( x − 4 + 6 − x ) is (a) (c) [4, 6] (2, 3)
f (x) =
3
(b) (d)
(–∞, 6) none of these
35.
Let (a) (c)
sin x 1 + sin x . If D is the domain of f, then D contains
(0, π) (3π, 4π)
(b)
(–2π, –π) (d) (4π, 6π)
36.
⎡ ( x + 1)(x − 3) ⎤ f (x) = ⎢ ⎥ ⎣ ( x − 2) ⎦ is a real valued function in the domain
(a) (b)
]–∞, –1] ∪ [3, ∞[ ]–∞, –1] ∪ ]2, 3]
37.
(c) ]–1, +2] ∪ [3, ∞[ (d) None of these If (x + 2y, x – 2y) = xy, then f(x, y) equals (a)
x 2 − y2 8 (b) x 2 − y2 x 2 + y2 4 (c) 4
(d)
x 2 − y2 2
38.
⎛1⎞ ⎛1⎞ f ( x )f ⎜ ⎟ = f ( x ) + f ⎜ ⎟ ⎝x⎠ ⎝ x ⎠ . If f(10) = A polynomial functions f(x) satisfies the condition
1001, then f(20) = (a) 2002 (c) 8001
(b) (d)
8008 none of these
39.
⎛ π2 ⎞ − x2 ⎟ f ( x ) = 3 sin ⎜ ⎜ 16 ⎟ ⎝ ⎠ lies in the interval The value of the function
⎡ π π⎤ ⎢− 4 , 4 ⎥ ⎣ ⎦
(a) (c) Solutions:
(–3, 3)
(b) (d)
⎡ 3 ⎤ ⎢0, 2 ⎥ ⎣ ⎦
none of these
1. 7. 13. 19. 25. 31. 37.
b b a a c c a
2. 8. 14. 20. 26. 32. 38.
c b c b c a c
3. 9. 15. 21. 27. 33. 39.
d a c d a c b
4. 10. 16. 22. 28. 34.
b b b c b a
5. 11. 17. 23. 29. 35.
b b d c a a
6. 12. 18. 24. 30. 36.
a a c a b c
ANALYSIS 1. Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a The relation R is (a) not symmetric (b) transitive (c) a function (d) reflexive [Ans. : a] 2. The range of the function f(x) = 7 – xPx – 3 is {1, 2, 3, 4} {1, 2, 3} (b) {1, 2, 3, 4, 5, 6} (d) {1, 2, 3, 4, 5} relation on the set A = {1, 2, 3, 4}.
(a) (c) [Ans. : c] 3.
If f : R → S, defined by f(x) = sin x – √3 cos x + 1, is onto, then the interval of S is (a) [0, 1] (b) [–1, 1] (c) [0, 3] (d) [–1, 3] [Ans. : d] 4. The graph of the function y = f(x) is symmetrical about the line x = 2, then (a) f(x) = f(–x) (b) f(2 + x) = f(2 – x) (c) f(x + 2) = f(x – 2) (d) f(x) = –f (–x) [Ans. : b]
f (x) = sin −1 ( x − 3) 9 − x2
The domain of the function (a) [1, 2] (b) [2, 3) (c) [2, 3] (d) [1, 2) [Ans. : b]
5.
is
6.
Let R ≡ {(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)} be a relation on the set A = {3, 6, 9, 12}. The relation is (a) reflexive only (b) reflexive and transitive only (c) reflexive and symmetric only (d) an equivalence relation [Ans. : b] 7. Let f : (–1, 1) → B, be a function defined by and onto when B is the interval (a)
⎡ π⎞ ⎢0, 2 ⎟ ⎣ ⎠ ⎛ π π⎞ ⎜− , ⎟ ⎝ 2 2⎠
f ( x ) = tan −1
2x 1 − x 2 , then f is both one-one
(b)
⎛ π⎞ ⎜ 0, ⎟ ⎝ 2⎠ ⎡ π π⎤ ⎢− 2 , 2 ⎥ ⎣ ⎦
(d) (c) [Ans. : c] 8. A real valued function f(x) satisfies the functional equation f(x – y) = f(x) f(y) – f (a – x) f (a + y) where a is a given constant and f(0) = 1, f(2a – x) is equal to (a) f(x) (b) –f(x) (c) f(–x) (d) f(a) + f(a – x) [Ans. : b]