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Relations and Functions Test - 4

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Relations and Functions Test - 4

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Question 1
1 / -0

If f (x) =√-x, then domain of fof is

A
(−∞,∞)

B
(−∞,0)

C
(0,∞)

D
none of these

Solution

Explanation:
√f(x) is defined only if
f(x)will be non negative.
Hence ∴√−f(x)is defined only for 0.
Question 2
1 / -0

Which of the following is/are not defined at x = 0?
a. 1/x
b. x²/x
c. x³/x
d. x³/x²

A
only a

B
b, c and d

C
a, b and c

D
all of the above
Question 3
1 / -0

The function f (x) = x + cos x

A
assumes only positive values

B
is defined for all reals

C
assumes only negative values

D
none of these

Solution

f(x)=x;g(x)=cosx
Df=R
Dg=R
D_f+g= Df∩Dg
so,
Df+g=R
SO given function is defined for all reals
Question 4
1 / -0

Which of the following is the most appropriate way to represent the equality of ordered pair?

A
(a, b) = (c, d) iff a = b = c = d

B
(a, b) = (c, d) iff a = c and b = d

C
(a, b) = (c, d) iff a = c and a = d or b = c and b = d

D
(a, b) = (c, d) iff a = c and a = d and b = c and b = d
Question 5
1 / -0

Let x be any real, then [x + y] = [x] + [y] holds for

A
y ∈ I

B
y ∈ R, y ∈ Q.

C
y ∈ R

D
y ∈ Q

Solution

[x]=x ∀ x∈I
so,
[x+y]=[x]+[y] only if x and y ∈ I
Question 6
1 / -0

If f(x) and g(x) are two functions, then which of the following operations may increase the domain of the resulting function?

A
f(x) + g(x)

B
f(x) – g(x)

C
Both (1) and (2)

D
Neither (1) nor (2)
Question 7
1 / -0

Which of the following statements are false?

a. Every function is a relation, but every relation is not a function.
b. Every relation is a function, but every function is not a relation.
c. There is no relation between functions and relations.
d. Function is just another name of relation.

A
a, c and d

B
a, b and d

C
b, c and d

D
a, b and c
Question 8
1 / -0

Which of the following statements defines any relation R between two sets A and B most appropriately?

A
R = A B

B
R A B

C
R A B

D
R = {ab; a A and b B}
Question 9
1 / -0

If P and Q are two non-empty sets, then which of the following statements defines cartesian product most appropriately?

A
P Q = {(p , q) : p P, q Q}

B
P Q = {p q : p P, q Q}

C
P Q = {p q : p P, q Q}

D
P Q = {(q , p) : p P, q Q}
Question 10
1 / -0

Let A and B be two given sets, then in order to define a relation on these two sets which of the following must hold?

A
Relation can be defined between any two sets

B
One of them should be non-empty

C
Both of them should be non-empty

D
There must be some connection between these two sets
Question 11
1 / -0

If, f is a function from A to B, then for a A and c, d B, f(a) = c and also f(a) = d, then

A
c > d

B
c < d

C
c = d

D
c d
Question 12
1 / -0

Which of the following is/are graph(s) of f(x) = x²?

A

B

C

D
All of the above
Question 13
1 / -0

Which of the following graphs will give all the real numbers as its values?

A

B

C

D
Question 14
1 / -0

Which of the following relations is/are function(s)?

A
R = {(2, 1), (3, 1), (4, 2)}

B
R = {(2, 2), (2, 4), (3, 3), (4, 4)}

C
R = {(1, 2), (2, 3), (1, 4), (4, 5), (5, 6), (6, 7)}

D
All of the above
Question 15
1 / -0

If set A = {1, 4, 8, 2} and set B = {3, 6}, then which of the following is A B?

A
{3, 12, 24, 6, 6, 24, 48, 12}

B
{(1, 3), (4, 3), (8, 3), (2, 3), (1, 6), (4, 6), (8, 6), (2, 6)}

C
{3, 12, 24, 6, 48,12}

D
{(3, 1), (3, 4), (3, 8), (3, 2), (6, 1), (6, 4), (6, 8), (6, 2)}
Question 16
1 / -0

If there are p elements in set A and q elements in set B, then number of elements in A ´ B are always equal to

A
pq

B
p^q

C
p + q

D
None of these
Question 17
1 / -0

Which of the following is a relation from set A = {1, 6, 8} to set B = {3, 6, 9}?

A
R = {(1, 3) (1, 5)}

B
R = {(6, 6)}

C
R = {(8, 1), (8, 9)}

D
All of the above
Question 18
1 / -0

If f (x) = tan x, −π2

A
[- 1, 1]

B
[−π4,π4]

C
(- 1, 1)

D
(−π/2,π/2)

Solution

fog(x)=f(g(x))=f(√1−x²)
⇒fog(x)=tan√ 1−x²
and it is defined if
1−x²≥0
⇒x²≤1
⇒−1≤x≤1
Question 19
1 / -0

Which of the following statements are true?
a. Every real function has a graph.
b. Graph of a real function is a subset of R²_^.c. Graph of a quadratic polynomial is always a parabola.
d. The graph of a general polynomial does not have a break in it.

A
a, b and d

B
a, c and d

C
b, c and d

D
all of the above
Question 20
1 / -0

Which of the following statements is/are true?

A
If we draw a line parallel to y-axis, then it can cut the graph of a function at a maximum of 2 points.

B
If we draw a line parallel to y-axis, then it can cut the graph of a function at a maximum of 1 point.

C
If we draw a line parallel to x-axis, then it can cut the graph of a function at a maximum of 1 point.

D
All of the above
Question 21
1 / -0

Which of the following could be the graph of a polynomial?

A

B

C

D
Question 22
1 / -0

Let N be the set of natural numbers. Define a function f : N N by f (x) = 2x + 1. Using this definition, which of the following values cannot be obtained?

A
17

B
8

C

D
All of the above
Question 23
1 / -0

Which of the following statements can be used to define a function f, from a set A to set B?
a. A relation from set A to set B is said to be a function, if every element of set A has one and only one image in set B.
b. A relation from set A to set B is said to be a function, if every element of set A has a unique image in set B.
c. If, f is a function from A to B, then for a A and c, d B, then if f(a) = c and also f(b) = d, then c d.

A
a and b

B
a, b and c

C
a and c

D
only a
Question 24
1 / -0

The diagram given below shows that

A
f is a function from A to B

B
f is a one-one function from A to B

C
f is a bijection from A to B

D
f is not a function

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