1 Marks Questions

1. Find a and b if (a – 1, b + 5) = (2, 3)If A = {1,3,5}, B = {2,3} find : (Question-2, 3)

Ans. a = 3, b = –2

2. A × B

Ans. A × B = {(1,2), (1,3), (3,2), (3,3), (5,2), (5,3)}

3. B × A Let A = {1,2}, B = {2,3,4}, C = {4,5}, find (Question- 4,5)

Ans. B × A = { (2,1), (2,3), (2,5), (3,1), (3,3), (3,5)}

4. A × (B ∩ C)

Ans. {(1,4), (2,4)}

5. A × (B ∪ C)

Ans. {(1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5)}

6. If P = {1,3}, Q = {2,3,5}, find the number of relations from A to B

Ans. = 64

7. If A = {1,2,3,5} and B = {4,6,9}, R = {(x, y) : |x – y| is odd, x ∈ A, y ∈ B} Write R in roster form

Which of the following relations are functions. Give reason.

Ans. R = { (1,4), (1,6), (2,9), (3,4), (3,6), (5,4), (5,6)}

8. R = { (1,1), (2,2), (3,3), (4,4), (4,5)}

Ans. Not a function because 4 has two images.

9. R = { (2,1), (2,2), (2,3), (2,4)}

Ans. Not a function because 2 does not have a unique image.

10. R = { (1,2), (2,5), (3,8), (4,10), (5,12), (6,12)} Which of the following arrow diagrams represent a function? Why?

Ans. Function

11.

Ans. Function

12.

Let f and g be two real valued functions, defined by, f(x) = x2, g(x) = 3x +2.

Ans. Not a function

13. (f + g)(–2)

Ans. 0

14. (f – g)(1)

Ans. -4

15. (fg)(–1)

Ans. -1

16.

Ans. 0

17. If f(x) = x3, find the value of,

Ans. 31

18. Find the domain of the real function, f(×) =

Ans. (–∞, –2] ∪ [2, ∞)

19. Find the domain of the function,  f (×) = Find the range of the following functions, (Question- 20,21)

Ans. R – {2,3}

20. f (x) =

Ans. (–∞, –0] ∪ [1, ∞)

21. f(x) = + 2

Ans. [2,∞)

22. Find the domain of the relation, R = { (x, y) : x, y ϵ Z, xy = 4} Find the range of the following relations : (Question-23, 24)

Ans. {–4, –2, –1,1,2,4}

23. R = {(a,b) : a, b ϵ N and 2a + b = 10}

Ans. {2,4,6,8}

24.R =

Ans.

25.If the ordered Pairsand are equal, findand

Ans.

26.Andare two sets Then no. of relations ofhave.

Ans. 64

27.Let then Range of function

Ans.

28.A real function is defined by Then the Value of

Ans. -11

29.If and form the setsandare these two Cartesian products equal?

Ans. Given and by definition of cartesion product, we set

and

By definition of equality of ordered pains the pair is not equal to the pair therefore

30..Ifand are finite sets such that and find the number of relations fromto

Ans. Linen

the number of subsets of

then the number of subsets of

Since every subset of is a relation from A to B therefore the number of relations from A to B = 2^mk

31.Let be a function from z to z defined by for same integers a and b determine a and b.

Ans. Given

Since

Subtracting (i) from(ii) we set a=2

Substituting a=2 is (ii) we get 2+b=1

b = -1

Hence a = 2, b = -1

32.Express as the set of ordered pairs

Ans. Since and

Put

For anther values of we do not get

Hence the required set of ordered peutes is

33.If find

Ans.

34.Functionis defined byfind

Ans.

35.Letbe a linear function from intofind

Ans.

36.If the ordered pairsand are equal, find &

Ans.

37.Let andbe the relation, is one less than from tothen find domain and Range of

Ans. Given and is the relation ‘is one less than’ from to therefore

Domain of and range of

38.Letbe a relation from todefine by.

Is the following true implies

Ans. No; let As so but so

39.Letbe the set of natural numbers and the relationbe define inby =what is the domain, co domain and range of? Is this relation a function?

Ans. Given

Domain of co domain of and Range of is the set of even natural numbers.

Since every natural number has unique image therefore, the relation is a function.

40.Let and list the element of

Ans.

41.Let be the subset of defined by

. Is a function from Justify your answer

Ans. Is not a function from Q to Z

One element have two images

is not function

42.The function which maps temperature in Celsius into temperature in Fahrenheit is defined by

Ans.

43.If Prove that

Ans.

44.If and are two sets containingand elements respectively how many different relations can be defined fromto?

Ans.

4 Marks Questions

1. Let A = {1,2,3,4}, B = {1,4,9,16,25} and R be a relation defined from A to B as, R = {(x, y) : x ϵ A, y ϵ B and y = x²}

(a) Depict this relation using arrow diagram.

(b) Find domain of R.

(c) Find range of R.

(d) Write co-domain of R.

Ans.

(b) {1,2,3,4}

(c) {1,4,9,16}

(d) {1,4,9,16,25}

2. Let R = { (x, y) : x, y ϵ N and y = 2x} be a relation on N. Find :

(i) Domain

(ii) Codomain

(iii) Range

Is this relation a function from N to N

Ans. (i) N

(ii) N

(iii) Set of even natural numbers

yes, R is a function from N to N.

3. Find the domain and range of, f(x) = |2x – 3| – 3

Ans. Domain is R

Range is [–3, ∞)

4. Draw the graph of the Constant function, f : R ϵ R; f(x) = 2  x ϵ R. Also find its domain and range.

Ans. Domain = R

Range = {2}

5.Let then

(i) Find the domain and the range of R (ii) Write R as a set of ordered pairs.

Ans. (i)Given and

Put

for all other values of we do not get

Domain of and range of

(ii) as a set of ordered pairs can be written as

6.Let R be a relation from Q to Q defined by show that

Ans.

(i)

(ii)

(iii)

7.

Ans.

8.Find the domain and the range of the function Also find and the numbers which are associated with the number 43 m its range.

Ans.

For must be real number

must be a real number

Which is a real number for every

let

We know that for all

which are associated with the number in

9.If

Ans.

10.Find the domain and the range of the function

Ans.

11.Let a relation then

(i) write domain of R

(ii) write range of R

(iii) write R the set builder form

(iv) represent R by an arrow diagram

Ans. Given

(i) Domain of

(ii)Rang of

(iii)R in the builder from can be written as

(iv) The reaction R can be represented by the arrow diagram are shown.

12.Let and

(i) find

(ii) write R in roster form

(iii) write domain & range of R

(iv) represent R by an arrow diagram

Ans. (i)

(ii)

(iii)Domain of and range of

(iv)The relation R can be represented by the are arrow diagram are shown.

13.The cartesian product has a elements among which are found and find the set and the remaining elements of

Ans. Let

Given

Given and

Also and

This but

Therefore

The remaining elements of are

14.Find the domain and the range of the following functions

Ans. Given

For must be a real number

Must be a real number

For let

As

15.Let and be two real functions. Find the following functions

Ans. Given and we note that and so there functions have the same Domain

(i) for

(ii)for all

(iii) for all

(iv)

(v)

for all

16.Find the domain and the range of the following functions

Ans. (i)Given

For must be a real number

must be a real number

set of all real number except

For

Must be a real number

Set of all real number except

(ii)Given

For must be a real number

Must be a real number

For let

But for all

Multiply both sides by a positive real number

(iii)Given

For must be a real number

Must be a real number

Set of all real number except

For let

But for all

But

Multicity bath sides by a positive real number

Either or but

17.If and

find

Ans.

18.For non empty sets and prove that

Ans. First we assume that

Then and

This, when then

Conversely, Let and let be

Then, for same

19.Let be given fixed positive integer. let show that is an equivalence relation on Z.

Ans.

(i)

(ii)Let Then

is divisible by

is divisible by

is divisible

Then

So is symmetric.

(iii) Let and

is divisible by and is divisible by

is divisible by

is divisible by

and

So, is transitive this is reflexive symmetric and transitive Hence, is an equivalence relation and.

20.Let and let be the relation, is greater than from to Write as a set of ordered pairs. find domain and range

Ans.

Domain of R Range of R

21.Define modulus function Draw graph.

Ans. let for each then

we know that for all

dom and range set of non negative real number

Drawing the graph of modulus function defined by

We have

Scale: 5 small divisions = 1 unit

On a graph paper, we plot the points and

Join them successively to obtain the graph lines AO and OG, as show in the figure above.

22.Let Show that is function, while g is notfunction.

Ans. Each element in has a unique image under

But, and

So is not a function

23.Let and write how many subsets will have? List them.

Ans. 16 Subsets of have

24.Let and verify that

Ans.

Part-I

Part-II

25.Find the domain and the range of the relation defined by

Ans.

26.Find the linear relation between the components of the ordered pairs of the relation where

Ans. Given

Let be the linear relation between the components of

Since

Also

Subtracting from , we get

Subtracting is , we get

Subtracting there values of a and b in we get

which is the required linear relation between the components of the given relation.

27.Let define a relation from toby

(i) write in the roaster form

(ii) write down the domain, co-domain and range of

(iii) Represent by an arrow diagram

Ans. (i)

(ii) Domain co domain range

(iii)

28.A relation is defined by where

(i) list the elements of

(ii) is a function?

Ans.

(i)

(ii)We note that each element of the domain of has a unique image; therefore, the relationis a function.

29.If Prove that

Ans.

30.Letbe defined by for all where and write the relation in the roster farm. It a function?

Ans.

is a function because different elements of have different imager in y

31.Determine a quadratic function defined by

Ans.

Multiplying eq. (i) by 3 and eq. (ii) by 2

32.Find the domain and the range of the function defied by

Ans.

For Df , must be a real no.

Domain of = set of all real numbers

33.Find the domain and the range of

Ans.

34. If

Ans.
(i)

(ii)

6 Marks Questions

1.Draw the graphs of the following real functions and hence find their range

Ans. Given

Let

(Fig for Answer 11)

Plot the points shown is the above table and join there points by a free hand drawing.

Portion of the graph are shown the right margin

From the graph, it is clear that

This function is called reciprocal function.

2.If Prove that

Ans. Ifprove that

3.Draw the graphs of the following real functions and hence find their range

Ans. (i)Given, which is first degree equation in and hence it represents a straight line. Two points are sufficient to determine straight lint uniquely

Table of values

A portion of the graph is shown in the figure from the graph, it is clear that y takes all real values. It therefore that

(ii)Given

Let

i.e which is a first degree equation is and hence it represents a straight line. Two points are sufficient to determine a straight line uniquely

Table of values

A portion of the graph is shown is the figure from the graph it is clear that y takes all real values except 2. It fallows that

4.Let f be a function defined by

(i) find the image of 3 under

(ii) find

(iii) find such that

Ans. Given

(i)

(ii)

(iii)

5.The functionis the formula to connect to Fahrenheit

units find interpret the result is each case

Ans.

6.Draw the graph of the greatest integer function,

Ans. Clearly, we have

7.Find the domain and the range of the following functions:

Ans. (i)Given

For must be a real number

Must be a real number

either

For let

As square root of a real number is always non-negative,

On squaring (i), we get

but for all

which is true for all also

(ii)Given

For must be a real number

must be a real number

For let

As square root of real number is always non-negative,

Squaring we get

but for all

but

(iii)Given

For must be a real number

must be a real number

For let

Also as the square root of a real number is always non-negative,

on squaring we get

But for all

(Multiply bath sides by a positive real number)

either or

8.Draw the graphs of the following real functions and hence find range:

Ans.

Given

Let

Plot the points

And join these points by a free hand drawing. A portion of the graph is shown in sigma (next)

From the graph, it is clear that takes all non-negative real values, if follows that

9.Define polynomial function. Draw the graph of find domain and range

Ans. A function define by

And is non negative integer is called polynomial function

Graph of

Domain of

Range of

10.(a) If are two sets such that and some elements of are than find

(b) Find domain of the function

Ans. (a)Given A and B are two sets such that

Some elements of are

(b)