## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 2 Relations and Functions

**Short Answer** **Type Questions**

**(Score 3)**

Question 1.

R is the relation “greater than” from A to B where A = {2,3,4,5,6} and B = {2,5,6} .

i. Write down the Roster form corresponding to R.

ii. Find the inverse relation to R.

Answer:

i.R= {(6,2), (5,2), (4,2), (3,2), (6,5)}

R^{-1 } ={(2,6), (2,5), (2,4), (2,3), (5,6)}

Question 2.

If A= {1,4,9} and B = {2,4,6}, find

i. (A∩B)×(A∪B)

ii. (A×B) ∩ (B×A)

Answer:

i. A ∩ B={4},

A∪B= {1,2,4,6,9}

(A∩B)×(A∪B)

= {(4,1), (4,2), (4,4), (4,6), (4,9)}

ii (A×B)∩ (B×A)={(4,4)}

Question 3.

If A= {1,2,3}, B = {2,4}, C = {72,5}

i. Find (B – C), (A × B), (A × C)

ii. Verify that A × (B – C)=(A ∪ B) – (A × C)

Answer:

i. B – C = {4}

A×B = {(1,2), (2,2), (3,2), (1,4), (2,4), (3,4)}

A×C = {(1,2), (2,2), (3,2), (1,5),(2,5), (3,5)}

ii. A×(B-C)={(l,4),(2,4),(3,4)}

(A×B)-(A×C)= {(1,4), (2,4), (3,4)}

= A×(B – C)

Question 4.

Range of the function

a. [0,1]

b.[1, ∞]

c. [1,0]

d. (1,-∞)

Answer:

Question 5.

a. If A= {x ∈ R: x^{2} – 5x + 6 = 0} and B = {x ∈ R: x^{2} = 9}, find AxB.

b. Let A= {1,2,3} and B = {2,4}. Find A×B and show it graphically..

Answer:

a.A={2,3}, B={-3,+3}

A×B ={(2,-3), (2,3), (3,-3), (3,3)}

b. A×B = {(1,2), (1,4), (2,2), (2,4), (3,2), (3,4)}

Question 6.

Let A = {1,2,3,4} and B = {5,7,9}. Determine

i. Is A×B = B×A?

ii. Is n(A×B) = n(B×A)?

Answer:

i. No. Since A × B and B × A do not have exactly the same ordered pairs.

∴ A×B ≠ B×A

ii. Yes.

n(A × B) = n(A). n(B) = 4 × 3 = 12

n(B × A) = n(B).n(A) = 3 × 4=12 n(A×B)

∴ n(B × A) =n(B×A)

Question 7.

If R is a relation on the set N of natural numbers defined by a + 3b -12, find

i. R

ii. Domain of R

iii. Range of R

Answer:

i. Given R={(a, b)/a, b∈N, a + 3b = 12}

∴R = {(9,1), (6,2), (3,3)}

ii. Domain of R= {9,6,3}

iii. Range of R= {1,2,3}

Question 8.

Given that f (x) = 2x^{2} – 3x

i. Find f(1) and f(2)

ii. What is the value of f (x + 2)

iii. Solve if f(x)=0

Answer:

i. f(1) = -1,f(2) = 2

ii. f(x + 2) = 2(x + 2)^{2}-3(x + 2) =2x^{2} + 5x + 2

iii. 2x^{2} – 3x = 0 ⇒ x = 0,^{3}/_{2}

Question 9.

Represent the relation {(x, x^{2} +1):x ∈N and 2 < x < 4} using an arrow diagram. Find its domain and range, h R is a relation “less than” from A to B where A={1, 2,3,4, 5} and B ={1,4,5}. Write down the relation R and R^{-1 }Answer:

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

R^{-1} ={(4, 1),(5, 1),(4, 2)45, 2), (4, 3), (5,3),(5,4)}

Question 10.

Consider the function f:R→R defined by

iii.Draw the graph of f(x)

Answer:

Question 11.

Let f and g be two function defined by and g(x) = .find

i. f+g

ii. f-g

iii. fg

iv.

Answer:

**Short Answer** **Type Questions**

**(Score 4)**

Question 1.

a. Let f be the relation on the set N of natural numbers defined by f = {(n, 3n): n ∈ N}. Is f a function from N to N? If so, find the range of f.

b. Find the maximum possible domain of

if it is a function.

Answer:

a. Since for each n∈N, there exists a unique 3n ∈N such that (n, 3n)∈ f.f is a function.

∴ Range of f={f(x):x∈N} = {3n/n∈N}

b. Domain of f = R – {1},since when

is not defined

Question 2.

a. Draw the graph of f: R→R such that f(x) = 4 – 2x.

b. Let A= {1,2,3,4}. Consider the following relations in A,

R_{1}= {(1,2), (2,3), (3,4), (4,1)}

R_{2} = {(1,1), (1,2), (1,3), (1,4)}

R3 = {(1,1), (2,1), (3,1), (4,1)}

R_{4}^{=}{(1,2), (3,4), (4,1)}

State whether or not each of these relations is a function of A into A

Answer:

b. R_{1 }is a function.

R_{2} is not a function, since 1 ∈ A and 1 appears as the first element in (1, 1) ∈ R_{2 }and (1,2) ∈ R_{2 }R_{3 }is a function.

R_{4} is not a function since 2 e A, but 2 does not appear as the first element in any ordered pair in R_{4}

Question 3.

Consider the function f: R→R defined by f(x) = |x-1|

i. Is f(2) = f(0), Why?

ii. Draw the graph of f(x).

iii. If g(x) = |x +1|, find (f+ g) and (f – g)

Answer:

i. Yes. f(2) = f(0)= 1

ii.

iii. (f+g) (2) = f (2) + g (2) = 1 + 3 = 4

(f – g) (2) = f (2) – g (2) = -2

Question 4.

Consider A = {1,2,3}, B = {4,5,6} and C = {1,5,6}

i. Find A×B and A×C

ii. B∩C =

iii. Verify A×(B∩C)=(A×B)∩(A×C)

Answer:

i. A×B = {(1,4), (1,5), (1,6), (2, 4), (2, 5), (2,6), (3,4), (3,5),(3,6)}

A×C= {(1,1),(1,5),(1,6),(2,1),(2,6), (3,1), (3,5), (3,6)}

ii. B∩C={5,6}

iii. A×(B∩C)={(1,-5),(1,6), (2,5),(2,6),(3,6)} = (A × B) ∩ (A × C)

Question 5.

a Determine the domain and range of the relation R defined R = {(x + 1, x + 6): x ∈ {0,1,2,3,4,5}. Find the domain and range of the function . where i∈R.

Answer:

a. R= {(1,6),(2,7),(3,’8),(4,9),(5,10), (6,11)}

Domain of R= {1,2,3,4,5,6}

Range of R= {6,7,8,9,10,11}

b.;f(1) and f (-1) are not defined.

Domain off = R – {-1,1}

Range = R-{0}

Question 6.

Let A = {-3, 2, 0, -1, 4}, B = {0, 1} and C ={4,3,-1}

i. Find A×B and A×C

ii. Find B ∪ C

iii. Verify A × (B∪C) = (A x B) ∪ (A x C)

Answer:

i. A × B = {(-3,0), (-3,1), (2,0), (2,1), (0,0)

(0,1), (-1,0), (1,1), (4,0), (4,1)}

A×C = {(-3,4), (-3,3), (-3, -1), (2,4), (2,3), (2,-1), (0,4), (0,3),(0,-1), (-1,4), (-1,3), (-1,-1), (4,4), (4,3), (4,-1)}

ii. B∪C={0,1,4,3,-1}

iii. A×(B∪C)= {-3,2,0,-1,4} × {0,1,4,3,-1} (A×B) ∪ (A× C) = A× (B ∪ C)

Question 7.

Consider two sets A = {1,2}, B ={3,4}

i. Write A × B.

ii. How many subsets will A x B have?

iii. How many relations possible from A to B?

iv. Define a relation R from A to B by R = {(x, y): y = x +1}. Write down the domain, co-domain and range of R.

Answer:

i. A × B = {(1,3), (1,4), (2,3), (2,4)}

ii. 2^{4}= 16 subsets

iii. 16 relations

iv. Domain={2}, co-domain={3,4},Range = {3}

Question 8.

a. If A=[a,b],B=[c,d],C = [d,e]then {(a, c),(a, d), (a, e), (b, c), (b, d), (b, e)}= ……………….

a. A∩(B∪C)

b. A∪(B∩C)

c. A ×(B∪C)

d. A × (B∩C)

b. Domain of the function

a. (1, ∞ )

b. (- ∞,4)

c. [-4, 4]

d. (-4, 0)

Answer:

a. A = [a, b] B = [c,d] C = [d, e] B∪C = [c, d, e]

A×(B∪C)= [a,b]× [c,d,e]

= {(a, c), (a, d), (a, e), (b, c), (b, d), (b, e)}

b. 16-x^{2} ≥0

x^{2} -16 ≤ 0 (x-4) (x + 4)≤ 0

-4<x<4

domain [-4,4]

Question 9.

Consider the real function

a. Find the domain and range of the function.

b. Prove that f(x).f(-x) + f(0) = 0

Answer:

a. Domain of the real function is given by x-2

i.e., Domain is R- {2}

Range of the function is R.

Question 10.

Which of the following are function ? Give reasons. If a function determine its domain and range.

i. {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

ii. {(0,0), (1,1), (1, -1), (4,2), (9,3), (9, -3), (16, -4)}

Answer:

i {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}

Here no two ordered pairs have the same first element, therefore it is a function. Further the function relates all elements in the domain {2,5,8,11,14,17} with a single 1 in the co-domain.

.’. f is a constant function and range of f = {1}.

ii. {(0,0), (1,1), (1,-1), (4,2), (9,3), (9, -3)

Here we see that the ordered pairs (1,1), (1,-1) have the same first element 1. So the given relation is not a function.

**Long Answer Type questions **

**(Score 6)**

Question 1.

a. Draw the graph of the function f(x) = |x|. Also write its domain and range,

b. Consider the following graphs.

i. Identify the graph of the funcgion f(x) =|x – 1|.

ii. Which of the above figures represent functions with range [1, ∞)?

iii. If the origin is shifted to (1,0), what is the equation of the curve in figure(i)?

Answer:

Range =[0,∞]

b.

i. Figure I

ii. Figure III

iii.|x|

Question 2.

Find the range of each of the following functions.

i. f(x) = 2 – 3x, x ∈ R , x > 0.

ii. f(x) = x^{2} + 2, x is a real number.

iii. f(x) = x, x is a real number

Answer:

i. Here,f(x) = 2-3x

Since, x∈R and x > 0.

3x >0 ⇒ -3x<0

⇒ 2-3x<2

∴ Range of function = {a∈R: a<2} = (- ∞,2)

ii. Here, f(x) = x^{2}+2

Since, x ∈ R

∴ x^{2}>0 for all x ∈ R⇒ x^{2} +2 > 2 Range of function

= {a∈R,a > 2, ∀ a ∈ R} = [2,∞ )

iii. Here,f(x) = x Since, x ∈R

∴ Range of function = R

Question 3.

Find domain and range of the real function f(x) defined by

Answer:

For x>0, x-1 >-1 and for x<0, -x>0 ⇒ 1 -x> 1

Hence, for x ∈ R, f(x)>-1

Domain = Set of real numbers;

Range = Set of real numbers > -1

Some points on graph are

Question 4.

a. Given, set A = {honest, violence} and B = {peace, prosperity, destruction}. Write the set A ×B, choose one element of A×B. Which values would you like to have in your life ?

b. Suppose a set A= {January, February, August} and set B = {28,15,30}. Write a relation R given by R= {(a, b)∈ A×B, where a is month and b has number of days}. Also, find R^{-1}.Which ordered pair represent an independence day ?

Answer:

a. Given, A = {honest, violence} and B = {peace, prosperity, destruction} A x B = {(honest, peace), (honest, prosperity), (honest, destruction), (violence, peace), (violence, property), (violence, destruction)}

I would like to have (honest, peace) values in my life,

b. Given sets are

A= {January, February, August} and B= {28,15,30}

R= {January 28), (January 15), (January 30), (February 28), (February 15), (February 30), (August 28), (August 15), (August 30)}

and R^{1} = {(January 28), (January 15), (January 30), (February 28), (February 15), (February 30),(August 28), (August 15), (August 30)} . The ordered pair which represent Independence day is (August, 15)

Question 5.

a. The largest relation possible from the set A = {0,1,2,4} to the set B={3,5} is…….

b. A relation from set of roll numbers of students in your class to the set of English alphabets, where relation is that each roll number a signed to first English alphabet of the name of the corresponding student. Is this relation a function? Justify your answer.

Answer:

a. Large relation is A ×B

a. Yes, It is a function. It satisfies all the conditions of a function.

Question 6.

Consider A = {1,2,3,…., 10}

i. n(A×A) =………………..

ii. Define a relation on A by

R={(x, y)/x+2y = 10, x, y ∈ A}. Express R in roster form.

iii. Is R a function on A? Justify your answer.

Answer:

i. n(A×A)= 10^{2}

ii. R= {(2,4),(4,3),(6,2), (8,1)}

iii. No, domain of R ≠ A

Question 7.

a. Find the range of the function {(2,1), (5,1), (8,1), (11,1), (14,1), (17,1)}.

b. Find the domain of the function f(x)=

c. The cartesian product A ×A has 9 elements among which two elements are (-1,0) and (0,1). Find the set A.

Answer:

a. {1}

b. [1-6]

x+1≥0 6-x≥0

x≥-1 x≤6

∴ domain [-1,6]

A={-1,0,1}

**NCERT Questions and Answers**

Question 1.

let be a real function from R to R. Determine the domain and range off.

Answer:

Question 2.

Consider the function

i. Fill up the following table:

ii. Find the domain and range of the function f.

iii. Sketch the graph of the above function.

Answer:

Question 3.

.

Answer:

Question 4.

Let A = {2, 3, 4, 5, 6} and R be the relation in A defined by {(a, b), a eA, b e A, a divides b}. Find

i. R

ii. Domain of R

iii. Range of R.

Answer:

i. We observe that

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

ii. Domain of R= {2,3,4,5,6}

iii. Range of R= {2,3,4,5,6}

Question 5.

Find the domain and range of the function

Answer:

D (f) = {x / f (x) is defined} =[-3,3]

Domain = [-3,3]

Range = R

Question 6.

Let._f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f (x) = ax + b, for some integers a, b. Determine a, b.

Answer:

( 1, 1)e.m.f(l)= 1 and (2,3) e f

⇒ f(2) = 3; .

a + b = 1 and

2a+b=3

∴ a=2,b=-1