QUESTION BANK
1. Let f : {1,3, 4} → {3, 4, 5} and g : {3, 4, 5} → {6,8,10} be function defined by f(1) =3,
f(3) = 4,f(4) = 5,g(3) = 6,g(4) = 8,g(5) = 8. Find gof(3)
2. Give an example of relation on a set A={1,2,3,4} which is reflexive and symmetric but
not transitive.
x
3. Show that f : [ −1,1] → R given by f ( x ) =
, is one-one.
x+2
4. Give an example of a relation, which is symmetric but neither reflexive nor transitive
5. Let R = , ∶ ℎ 5 be a relation.Find the range of R
6. Show that the relation R in the set of real numbers defined as = , : ≤ is
reflexive and transitive but not symmetric
7. Show that the relation R in the set = 1, 2, 3,4,5 is given by = , : | −
|
", is an equivalence relation.
8. Show that the function f : → defined as #$ = $ % is neither one-one nor onto.
9. Let
f : {1, 2, 3, 4} → {5}
defined by f={(1,5),(2,5),(3,5),(4,5)}. Does the function
invertible?
7 
3
10. A = R −   , B = R −  
5
5
f : A → B defined by f ( x ) =
3x + 4
, g : B → A defined by
5x − 7
7y + 4
,Find gof.
5y − 3
11. Let N be the set of natural numbers and f be the function f : N → N defined by
g ( y) =
x −1 if x >1
f(x) = x+1, x∈N. Prove that f is not onto function.If g(x) = 
then find gof.
if x =1
1
Check gof is an onto function
12. If f : R → R given by f ( x ) = x 2 − 3 x + 2 , find fof(x) and fof(1).
13. Consider f : R → R
given by
f ( x) = 5 x + 2 . a) Show that f is one-one. b) Is f
invertible? Justify your answer
14. If f : R → R g : R → R
defined by
f ( x ) = x 2 , g ( x) = 2 x + 3 i) Find fog.
that fog ≠ gof iii) Is f one-one? Justify your answer.
15. If #$ = 3$ − 1, &$ = $ % + 1 . Find fog, gof and fof
16. Let f : R → R given by f ( x) =
2x +1
, find fof and show that f is invertible.
3
ii)Show
17. Show that the function f : → , f(x) = 2x + 3 is invertible and find # )*
-
18. Find &+# and #+&, if #$ = 8$ and &$ = $ .
19. Let f : → defined by f(x)= 3x – 7.Show that f is invertible. Find # )*
20. Let f ( x) =
x −1
x−3
, x ≠ 3 and g ( x ) =
, x ≠1
x−3
x −1
be two real valued function a) Find
fog(x) b)Find f-1(x) and g-1(x) c) Find (gof)-1(x)
21. Let ∗ be a binary operation on Q defined by a ∗ b =
/.0
1
, Find the inverse of 9 with
respect to *.
22. Let * be binary operation on N defined by a * b = HCF of a and b. i) Is ∗ commutative. ii)
Is ∗associative.
23. Consider the set A={1,2,3,4,5}, B={1,4,9,16,25} and a function f : A → B
given by
f(1)=1, f(2)=4,f(3)=9,f(4)=16 and f(5)=25 a) Show that f is one-one. b) Show that f is
on-to. C) Does f-1 exist.
24. Let ∗ be a binary operation on Q defined by a ∗ b = ab + 1. Determine whether ∗ is
commutative or associative or not.
25. Let A = N x N and ∗ be the binary operation on A defined by (a , b) ∗ (c , d) =
(a + c , b + d). Show that ∗ is commutative and associative. Find the identity element for
∗ on A, if any.
26. Let Q be the set of rational numbers and ∗ be the binary operation on Q defined by
a∗b=
/.0
2
for all a,b in Q a) What is the identity element of * on Q b)Find the inverse
element of ‘a’under ∗ on Q. c)Show that (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a,b,c in Q.
27. Let A = N x N and ∗ be a binary operation on A defined by
(a, b) * (c, d) = (ac - bd, ad + bc). Show that ∗ is commutative on A.
28. Let ∗ : R x R → R given by a ∗b = 3% − Find the value of 2∗ 3. Is ∗ commutative.
Justify you
29. Let * be a binary operation on Q defined by a ∗ b =
/.0
Check whether ∗ is commutative
and associative.
30. Let * be a binary operation defined on the set of natural numbers by a ∗ b = a+b-1
i) Verify ∗ is associative. ii) Find the identity element if it exist.