MATHEMATICS
Daily Practice Problems
Target IIT-JEE 2024
TOPIC - FUNCTION DPP. NO.-13
1 2
Q.1 Which is the inverse of the function f (x) = ln x x 1 ?
3
1 3x 1 –3x 3x 1 3x –3x
(A) 3(e3x + e–3x) (B) (e + e–3x) (C) (e – e ) (D*) (e – e )
3 2 2
Q.2 Let a and be the roots of x2 – 6x – 2 = 0, with a > b. If an = n – n for n 1, then the
a10 2a 8
value of is
2a 9
(A) 1 (B) 2 (C*) 3 (D) 4
2009
k f 4 (k )
Q.3 Let f (k) =
2009
and g(k) =
(1 f ( k )) 4 (f ( k )) 4
then the sum g(k ) is equal :
k 0
(A) 2009 (B) 2008 (C*) 1005 (D) 1004
x lnx
Q.4 f (x) = and g (x) = . Then identify the CORRECT statement
lnx x
1 1
(A*) and f (x) are identical functions (B) and g(x) are identical functions
g(x) f (x )
1
(C) f (x) . g (x) = 1 x 0 (D) 1 x 0
f ( x ) . g (x )
ax 8 bx 6 cx 4 dx 2 15x 1
Q.5 Suppose that f (x) is a function of the form f (x) = (x 0). If
x
f (5) = 2 then the value of f (– 5) is equal to
(A) – 2 (B*) 28 (C) 13 (D) – 13
4 4 4x
Q.6 Let f : R R be a function defined as f(x) = . The inverse of f is the
3 3 3x 4
4 4
map g : R – R – is given by (4 – 3y) x = 4y
3 3
3y 4y 4y 3y
(A) g(y) = (B*) g(y) = (C) g(y) = (D) g(y) =
3 4y 4 3y 3 4y 4 3y
2x
Q.7 A function f : R R, f(x) = is
1 x2
(A) injective by not surjective (B) surjective but not injective
(C) injective as well as surjective (D*) neither injective nor surjective
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2
Q.8 M J Let f (x) = (3x + 2)2 – 1, – < x . If g(x) is the function whose graph is the reflection of the
3
graph of f(x) with respect to line y = x, then g(x) equals
1 1
(A*)
3
2 x 1 , x 1 (B)
3
2 x 1 , x 1
1 1
(C)
3
1 x 2 , x 2 (D)
3
1 x 2 , x 2
1
1 7
Q.9 If g(x) = 4 cos 4 x 2 cos 2 x cos 4x x 7 , then the value of gg (100) is equal to
2
(A) – 1 (B) 0 (C) 1 (D) 100
Q.10 Identify the correct statement
(A) If f (x) is periodic and g(x) is aperiodic, then f (g (x) ) must be aperiodic
(B) If f (x) and g(x) are both aperiodic, then f (g (x) ) must be aperiodic
(C*) If f (x) is aperiodic and g(x) is periodic, then f (g(x) ) must be periodic
(D) If f (x) is periodic and g(x) is aperiodic, then g (f (x) ) must be aperiodic.
Q.11 A function f is defined for all positive integers and satisfies f(1) = 2005 and
f(1) + f(2) + ... + f(n) = n2f(n) for all n > 1. The value of f(2004) is
1 1 2004
(A*) (B) (C) (D) 2004
1002 2004 2005
Q.12 Let f (x) be a function with two properties
(a) for any two real number x and y, f (x + y) = x + f (y) and
(b) f (0) = 2.
The value of f (100), is
(A) 2 (B) 98 (C*) 102 (D) 100
Q.13 For x R, the function f (x) satisfies 2 f (x) + f (1 – x) = x2 then the value of f (4) is equal to
13 43 23
(A) (B) (C*) (D) none
3 3 3
[MULTIPLE OBJECTIVE TYPE]
x
Q.14 Consider the function f (x) = . Which of the following statements are correct?
x 1
(A*) f has the same domain and range. (B*) f has its own inverse.
(C) f is not injective (D*) f is neither odd nor even.
Q.15 If f (x) = x3 – x2 + 100x + 2002, then
1 1
(A) f (1000) > f (1001) (B*) f >f
2000 2001
(C*) f (x – 1) > f (x – 2) (D) f (2x – 3) > f (2x)
Q.16 Which of the following functions are aperiodic
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1
(A) f (x) = 2 + (–1)[x] (B*) f (x) = cosx
x
(C) f (x) = tan ( x [ x ]) (D*) f (x) = x + sin x
4
where [x] denotes greatest integer function.
[Hint: (A) period 2, (C) period 1 ]
Q.17 Which of the following functions have the same period?
1 1 1
(A*) f (x) = sin2x + cos4x + 2 (B*) g (x) = + where f (x)=
f (sin x ) f (cos x ) 1 x2
| sin x | | cos x |
(C) h (x) = (D*) k (x) = cos(cos x) + cos(sin x)
| sin x cos x |
Q.18 Let f : R R defined by f (x) = Min. ( | x |, 1–| x |)
Then which of the following hold(s) good?
(A) Range of f is (–, 1] (B*) f is aperiodic.
(C) f is neither even nor odd. (D*) f is neither injective nor surjective.
Q.19 Which of the following function is periodic?
(A*) f(x) = sgn(e–x) where sgn x denotes signum function of x.
(B*) f(x) = sin x + |sin x|
(C*) f(x) = min.(sin x, | x |)
1 1
(D*) f(x) = x + x + 2[–x] where [x] denotes greatest integer less than or equal to x.
2 2
Q.20 Let f : A B and g : B C be two functions and gof : A C is defined. Then which of the
following statement(s) is/are INCORRECT?
(A*) If gof is into then f must be into.
(B*) If f is into and g is onto then gof must be onto function.
(C*) If gof is one-one then g must be one-one.
(D*) If f is surjective and g is injective then gof must be surjective.
Q.21 Let f : A B and g : B C be functions and gof : A C. Which of the following
statements is true?
(A) If gof is one-one then f and g both are one-one.
(B*) If gof is one-one then f is one-one.
(C*) If gof is a bijection then f is one-one and g is onto.
(D*) If f and g are both one-one then gof is one-one.
Q.22 Which of the following pair of functions have the same graph?
[Note : [k], {k} and sgn k denote the largest integer less than or equal to k, fractional part of k
and signum function of k respectively.]
(A*) f (x) = ln 1 {x} and g (x) = ln 1 {x} ,
1 1
(B) f (x) = and g (x) = 2 sin2x + cos 2x
1 tan x 1 cot2 x
2
2
3 x 3)
(C*) f (x) = 2sgn( x and g(x) = 2 sgn (x2 + 3x + 3)
sgn x |x|
(D*) f (x) = and g (x) =
sgn x |x|
INTEGER PAGE # 3
Q.19 Let f : {x, y, z} [a, b, c] be a one-one function and 1 one of the conditions
only
Q.23 Given a function f (x) satisfying f (x) + 2 f = x. Find f (2). [Ans. 2/3]
(i) f(x) b (ii) f (y) = b xf(z) a
1(iii)
is true then the function f is given by the set
Q.24(a) Polynomial P(x)
{(x, a), (y, b), (z,contains
c)} (b) only
{(x, terms
a), (y, of
c),odd
(z, degree.
b)} When
(c)*{x,P(x)b),is(y,
divided
a), (z, by
c)}(x – 3), the
(d) remainder
{(x, c),
2
is 6. I f P(x) is divided by (x – 9) then remainder is g (x). Find the value of g (2). Ans. 4]
(y, b), (z, a)}
Q.19 Let f : {x, y, z} [a, b, c] be a one-one function and only 3 of the conditions
2 xone
Q.25 Let f be a real valued invertible function such that f = 5x – 2, x 2. Find f –1(13).
(i) f(x) b (ii) f (y) = b xf (z)2 a
(iii)
is true then the function f is given by the set [Ans. 0003 ]
(a) {(x, a), (y, b), (z, c)} (b) {(x, a), (y, c), (z, b)} (c)*{x, b), (y, a), (z, c)} (d) {(x, c),
(y, b), (z, a)}
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