Assignment 2
Instruction. Solve the following problems. Assume any data if necessary. Note the page
number of the uploaded textbook material for Unit I.
BATCH I : 2023001173 to 2023001899
1. Unit I pg 12 : 12
2. Unit I pg 35 : 27
3. Unit I pg 64 : 10, 30
4. Find the domain of the function f given by
f (x) =
√
1
x2 − x − 2 + √
.
3 + 2x − x2
5. Find the range of the function f given by
f (x) =
x4 + 1
x2
6. Construct a bijection from R to (−1, 1).
7. Define the following relation R on the xy-plane R2 :
R = {(p, q) ∈ R2 × R2 | distance between p and q is 1}.
Is R reflexive ? Is R symmetric ?
BATCH II : 202300202 to 2023002730
1. Unit I pg 12 : 14
2. Unit I pg 35 : 24
3. Unit I pg 64 : 11, 29
4. Find the domain of the function f given by
f (x) =
√
1
x2 − x + 2 + √
.
3 + 2x − x2
1
5. Let f : A −→ B and g : B −→ C be two functions. Prove the following.
(a) If g ◦ f is one-one, then so is f .
(b) If g ◦ f is onto, then so is g.
6. Is the following function f : R → R a bijection ?
f (x) = x2 + x + 1.
7. Define the following relation R on Z as follows :
R = {(a, b) ∈ Z2 | 5 divides a − b}.
Is R symmetric and transitive ?
BATCH III : 2023002831 to 2023002405
1. Unit I pg 12 : 35
2. Unit I pg 35 : 20
3. Unit I pg 64 : 7, 31
4. Find the domain of the function f given by
√
√
f (x) = x2 − x − 2 + 3 + 2x − x2 .
5. Find the range of the function f given by
1
2 + x2
f (x) =
6. Is the following function f : R → R a bijection ?
f (x) = tan πx/2
7. Find the number of symmetric relations on the set A = {x ∈ Z | |x2 − 1| < 100}.
BATCH IV : 2023003559 to 2023004005
1. Unit I pg 12 : 36
2. Unit I pg 35 : 16
3. Unit I pg 64 : 12, 28
4. Find the domain of the function f given by
f (x) = loge
2
x2 − 5x + 6
x2 + 4x + 6
5. Find the range of the function f given by
f (x) = sin 2x + cos 2x
6. Is the following function f : R → R a bijection ?
f (x) = x3 + 1.
7. Find the number of reflexive relations on the set A = {x ∈ Z | |x2 − 1| < 100}.
BATCH V : 2023004035 to 2023004336
1. Unit I pg 12 : 11
2. Unit I pg 35 : 15
3. Unit I pg 64 : 8, 27
4. Find the domain of the function f given by
f (x) = loge
x2 + 5x + 6
x2 + 4x + 6
5. Find the range of the function f given by
f (x) =
2
2 − cos 3x
6. Is the following function f : R → R a bijection ?
f (x) = ln sec x
7. Define the following relation R on the xy-plane R2 :
R = {(p, q) ∈ R2 ×R2 | distance between p and the origin = distance between q and the origin}.
Is R an equivalence relation ?
BATCH VI: 2023004346 to 2023004868
1. Unit I pg 12 : 32
2. Unit I pg 35 : 14
3. Unit I pg 64 : 9, 26
4. Find the domain of the function f given by
f (x) = loge
3
x2 − 5x + 6
x2 + 4x + 4
5. Find the range of the function f given by
f (x) =
x
1 + x2
6. Define the following relation R on the xy-plane R2 :
R = {(p, q) ∈ R2 ×R2 | distance between p and the origin = distance between q and the origin}.
Is R reflexive ? Is R symmetric ?
BATCH VII : 2023004871 to 2023005591
1. Unit I pg 12 : 33
2. Unit I pg 35 : 10
3. Unit I pg 64 : 13, 25
4. Find the domain of the function f given by
1
f (x) = p
[x] − x
5. Find the range of the function f given by
f (x) =
2
cos x − 2
6. Let f : R → R and g : R → R be bijections. Then are the functions f + g and f g
bijective ? ((f + g)(x) = f (x) + g(x) and (f g)(x) = f (x)g(x).)
7. Define the following relation R on the xy-plane R2 :
R = {(p, q) ∈ R2 × R2 | distance between p and q is 1}.
Is R an equivalence relation ?
BATCH VIII : 2023005834 to 2023006587
1. Unit I pg 12 : 13
2. Unit I pg 35 : 30
3. Unit I pg 64 : 19, 20
4. Find the domain of the function f given by
f (x) = log10
4
p
x − [x]
5. Find the range of the function f given by
f (x) =
2
2 − cos 3x
6. Construct a bijection between (−∞, 1] and (0, 1].
7. Define the following relation R on Z as follows :
R = {(a, b) ∈ Z2 | 5 divides a − b}.
Is R an equivalence relation ?
5