AHLCON PUBLIC SCHOOL
ASSIGNMENT -1
CLASS XII
(2014-2015)
TOPIC: RELATIONS AND FUNCTIONS
Q.1
Which one of the following graph represents an identity function? Why?
(a)
Q.2
(b)
Which one of the following graph represents a constant function? Why?
(a)
Q.3
Find the domain of f(x) =
(b)
1
x  [ x]
𝑎 + 𝑏 𝑖𝑓𝑎 + 𝑏 < 6
Define a binary operation * on the set A={0,1,2,3,4,5} as a*b={
show that 0 is the
𝑎 + 𝑏 − 6 𝑖𝑓𝑎 + 𝑏 ≥ 6
identity for this operation and each element a of the set is invertible with 6-a being the inverse of a.
Q5
Let Q be the set of rational numbers and R be the relation on R ={(𝑥, 𝑦): 1 + 𝑥𝑦 > 0}. Prove that R is
Q.4
reflexive and symmetric but not transitive.
Q.6
Show that the function f : R  R defined by f(x) = x2 is not bijective.
Q.7
Given A = {-1, 0, 2, 5, 6}, B = { -2, -1, 0, 18, 28 } and f (x) = x2 – x – 2. Show that function f is not onto.
Q.8
Show that the function f : N  N given by f(1) = f(2) = 1 and f(x) = x – 1 for every x > 2, is onto but
not one-one.
Q9
Show that the function f:R R defined by f(x) = 2x3-7 for all x R is bijective.
Q 10 Show that the function f:R R defined by f(x) = x3 + 3x for all x R is bijective.
2x
 3
Q.11 Let f:    R be a function defined as f ( x) 
, find f-1.
5
x

3
5
 
5x  3
, show that f  f (x) is an identity function.
4x  5
Q.12
If f ( x) 
Q.13
Let f : 2,   R and g :  2,   R be two real functions defined by f(x) =
x2 ,
g ( x)  x  2 .
Find f+g and f-g.
Q.14
4x
4
Let f : R     R be a function defined by f ( x) 
, Find f-1.
3x  4
3
Q.15
Show that the function f : R  R defined by f(x) =
2x 1
, x  R is one – one and onto function. Also
3
find the inverse of the function f.
Q.16
Consider f : R  R , given by f(x) = 7x +30. Show that f is invertible. Find the inverse of f.
.Q.17 If f(x) = x2 and g (x)=x+1 , show that fog  gof.
Q.18
Let
f : R  R be defined by f(x) = 10x + 7. Find the function g: R  R such that
gof = fog = IR
Q.19
1 x 
 2x 
If f ( x)  log 
= 2f (x).
 Show that f 
2 
1 x 
1 x 
Q.20
 1 
If f (x) = 
 , find
1 x 
Q.21
1
If f(x) =   , g(x) =
x
  1 
f  f   
  2 
x and h (x) = x 2  1 , find
i) fogoh (x)
ii) hofog (x)
Q 22 If f:R R is given by f(x) = 3x +2 f or all x R and g : R R is given by g(x)=
x R find a) fog b)gof
for all
c)fof d)gog
Q 23
If f: R R is given by f(x)=(5 – x5)1/5 then find fof.
Q24
If f(x) =
and g(x)=
for all x
a) (gof)(-5/3) b) (fog)(-5/3)
R ,find
Q 25 If f(x) = x+7 and g(x) = x-7 , find
a) (fog)(7) b) (gog)(7) c) (gof)(7)
d) (fof)(7)
then find f-1 :range of f R-{2}.
Q 26 If f: R-{2} R is a function defined by f(x) =
Q 27 If f:R-{-3/5}
Q.28
, then find f-1 :range of f R-{-3/5}.
R is a function defined by f(x) =
Examine which of the following is a binary operation:
i) a  b 
ab
; a, b  N
2
ii) a  b 
ab
; a, b  Q
2
For binary operation check the commulative and associative property.
Q.29
Let X be a non – empty set. P(x) be its power set. Let ‘*’ be an operation defined on elements of P(x) by,
A  B  A  B A, B  P(x). Then,
Q.30
i)
Prove that * is a binary operation in P(x)
ii)
Is * cumulative?
iii)
Is * associative?
iv)
Find the identity element in P(x).
Let A = Q x Q. Let * be a binary operation on A defined by (a, b) * (c, d) = (ac, ad + b) then find i) identity
element
ii) invertible elements of (A, *).
Q 31 Let A be the set of all real numbers except -1 . Let * defined on A as a*b = a+ b + ab for all a, b
R .
Prove that
a) * is binary.
b) The given operation is commutative and associative.
c) The number 0 is the identity.
d) Every element a of A has –a/(1+a) as inverse.
e) Solve the equation 2*x*5 = 4
Q32. Let A = {1,2,3,4….,9} and R be the relation in AxA defined by (a,B) R (c,d) if a+d = b+c for (a,b), (c,d) in
AXA. Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)].