# NCERT Exemplar Solutions for Class 11 Maths Chapter 2 Relations and Functions

NCERT Exemplar Solutions for Class 11 Maths Chapter 2 Relations and Functions are created by the expert faculty at BYJUâ€™S. These Solutions of NCERT Exemplar Maths help the students in solving the problems adroitly and efficiently. They also focus on cracking the Solutions of Maths in such a way that it is easy for the students to understand. The NCERT Exemplar Solutions for Class 11 aim at equipping the students with detailed and step-wise explanations for all the answers to the questions given in the exercises of this Chapter.

Chapter 2 Relations and Functions of NCERT Exemplar Solutions for Class 11 Maths explains the concepts related to relations and functions. A relation can simply be defined as a set of outputs as well as inputs. These are also written as ordered pairs by representing a relation of a graph or by a mapping diagram. The graph of the relationship is one of the best visual indicators between the inputs and the outputs. Functions, on the other hand, can simply be defined as a relation with a single input and output. These materials can also be used as a reference tool while preparing for final exams. Click here to get exemplar solutions for all the chapters. In Chapter 2, students will learn and solve exemplar problems based on topics like;

• Cartesian product of sets
• Relations between sets
• Functions on sets
• Identity function
• Constant function
• Polynomial function
• Rational function
• The Modulus function
• Signum function
• Greatest integer function
• Algebra of functions
• Addition of two real functions
• Subtraction of a real function from another
• Multiplication by a Scalar
• Multiplication of two real functions
• Quotient of two real function
• Graphical representation of functions
• Algebraic representation of real functions

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Exercise Page No: 27

1. Let A = {â€“1, 2, 3} and B = {1, 3}. Determine
(i) A Ã— B

(ii) B Ã— A
(iii) B Ã— B

(iv) A Ã— A

Solution:

According to the question,

A = {â€“1, 2, 3} and B = {1, 3}

(i) A Ã— B

{â€“1, 2, 3} Ã— {1, 3}

So, A Ã— B = {(â€“1, 1), (â€“1, 3), (2, 1), (2, 3), (3, 1), (3, 3)}

Hence, the Cartesian product = {(â€“1, 1), (â€“1, 3), (2, 1), (2, 3), (3, 1), (3, 3)}

(ii) B Ã— A.

{1, 3} Ã— {â€“1, 2, 3}

So, B Ã— A = {(1, â€“1), (1, 2), (1, 3), (3, â€“1), (3, 2), (3, 3)}

Hence, the Cartesian product = {(1, â€“1), (1, 2), (1, 3), (3, â€“1), (3, 2), (3, 3)}

(iii) B Ã— B

{1, 3} Ã—{1, 3}

So, B Ã— B = {(1, 1), (1, 3), (3, 1), (3, 3)}

Hence, the Cartesian product = {(1, 1), (1, 3), (3, 1), (3, 3)}

(iv) A Ã— A

{â€“1, 2, 3} Ã— {â€“1, 2, 3}

So, A Ã— A = {(â€“1, â€“1), (â€“1, 2), (â€“1, 3), (2, â€“1), (2, 2), (2, 3), (3, â€“1), (3, 2), (3, 3)}

Hence,

the Cartesian product ={(â€“1, â€“1), (â€“1, 2), (â€“1, 3), (2, â€“1), (2, 2), (2, 3), (3, â€“1), (3, 2), (3, 3)}

2. If P = {x : x < 3, xÂ âˆˆÂ N}, Q = {x : x â‰¤ 2, xÂ âˆˆÂ W}. Find (PÂ âˆªÂ Q) Ã— (PÂ âˆ©Â Q), whereÂ WÂ is the set of whole numbers.

Solution:

According to the question,

P = {x: x < 3, xÂ âˆˆN}, Q = {x : x â‰¤ 2, xÂ âˆˆW} whereÂ WÂ is the set of whole numbers

P = {1, 2}

Q = {0, 1, 2}

Now

(PâˆªQ) = {1, 2}âˆª{0, 1, 2} = {0, 1, 2}

And,

We need to find the Cartesian product of (PâˆªQ) = {0, 1, 2} and (Pâˆ©Q) = {1, 2}

So,

(PâˆªQ) Ã— (Pâˆ©Q) = {0, 1, 2} Ã— {1, 2}

= {(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)}

Hence, the Cartesian product = {(0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)}

3. If A = {x : xÂ âˆˆÂ W, x < 2}, B = {x : xÂ âˆˆÂ N, 1 < x < 5}, C = {3, 5} find
(i) A Ã— (BÂ âˆ©Â C)
(ii) A Ã— (BÂ âˆªÂ C)

Solution:

According to the question,

A = {x: xÂ âˆˆ W, x < 2}, B = {x : xÂ âˆˆN, 1 < x < 5} C = {3, 5};Â WÂ is the set of whole numbers

A = {x: xÂ âˆˆ W, x < 2} = {0, 1}

B = {x : xÂ âˆˆN, 1 < x < 5} = {2, 3, 4}

(i)

A Ã— (Bâˆ©C) = {0, 1} Ã— {3} = {(0, 3), (1, 3)}

Hence, the Cartesian product = {(0, 3), (1, 3)}

(ii)

(BâˆªC) = {2, 3, 4} âˆª {3, 5}

(BâˆªC) = {2, 3, 4, 5}

A Ã— (BâˆªC) = {0, 1} Ã— {2, 3, 4, 5} = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)}

Hence, the Cartesian product = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)}

4. In each of the following cases, find a and b.
(i) (2a + b, a â€“ b) = (8, 3)

(ii) (a/4 , a â€“ 2b) = (0, 6 + b)

Solution:

(i)

According to the question,

(2a + b, a â€“ b) = (8, 3)

Given the ordered pairs are equal, so corresponding elements will be equal.

Hence,

2a + b = 8 and aâ€“b = 3

Now aâ€“b = 3

â‡’a = 3 + b

Substituting the value of a in the equation 2a + b = 8,

We get,

2(3 + b) + b = 8

â‡’Â 6 + 2b + b = 8

â‡’Â 3b = 8â€“6 = 2

â‡’ b = 2/3

Substituting the value of b in equation (aâ€“b = 3),

We get,

â‡’ a â€“ 2/3 = 3

â‡’ a = 3 + 2/3

â‡’ a = (9 + 2)/3

â‡’ a = 11/3

Hence the value of a = 11/3 and b =Â 2/3Â respectively.

(ii)

According to the question,

Given the ordered pairs are equal, so corresponding elements will be equal.

a/4 = 0Â and a â€“ 2b = 6 + b

NowÂ a/4 = 0

â‡’a = 0

Substituting the value of a in the equation (aâ€“2b = 6 + b),

We get,

0 â€“ 2b = 6 + b

â‡’Â â€“ 2b â€“ b = 6

â‡’Â â€“ 3b = 6

â‡’ b = â€“ 6/3

â‡’ b = â€“ 2

Hence, the value of a = 0 and b = â€“ 2 respectively

5. Given A = {1, 2, 3, 4, 5}, S = {(x, y) : xÂ âˆˆÂ A, yÂ âˆˆÂ A}. Find the ordered pairs which satisfy the conditions given below:

(i) x + y = 5

(ii) x + y < 5

(iii) x + y > 8

Solution:

According to the question, A = {1, 2, 3, 4, 5}, S = {(x, y) : xÂ âˆˆA, yÂ âˆˆA}

(i) x + y = 5

So, we find the ordered pair such that x + y = 5, where x and y belongs to set A = {1, 2, 3, 4, 5},

1 + 1 = 2â‰ 5

1 + 2 = 3â‰ 5

1 + 3 = 4â‰ 5

1 + 4 = 5â‡’Â the ordered pair is (1, 4)

1 + 5 = 6â‰ 5

2 + 1 = 3â‰ 5

2 + 2 = 4â‰ 5

2 + 3 = 5â‡’Â the ordered pair is (2, 3)

2 + 4 = 6â‰ 5

2 + 5 = 7â‰ 5

3 + 1 = 4â‰ 5

3 + 2 = 5â‡’Â the ordered pair is (3, 2)

3 + 3 = 6â‰ 5

3 + 4 = 7â‰ 5

3 + 5 = 8â‰ 5

4 + 1 = 5â‡’Â the ordered pair is (4, 1)

4 + 2 = 6â‰ 5

4 + 3 = 7â‰ 5

4 + 4 = 8â‰ 5

4 + 5 = 9â‰ 5

5 + 1 = 6â‰ 5

5 + 2 = 7â‰ 5

5 + 3 = 8â‰ 5

5 + 4 = 9â‰ 5

5 + 5 = 10â‰ 5

Therefore, the set of ordered pairs satisfying x + y = 5 = {(1,4), (2,3), (3,2), (4,1)}.

(ii) x + y < 5

So, we find the ordered pair such that x + y<5, where x and y belongs to set A = {1, 2, 3, 4, 5}

1 + 1 = 2<5 â‡’ the ordered pairs is (1, 1)

1 + 2 = 3<5 â‡’ the ordered pairs is (1, 2)

1 + 3 = 4<5 â‡’ the ordered pairs is (1, 3)

1 + 4 = 5

1 + 5 = 6>5

2 + 1 = 3<5 â‡’ the ordered pairs is (2, 1)

2 + 2 = 4<5 â‡’ the ordered pairs is (2, 2)

2 + 3 = 5

2 + 4 = 6>5

2 + 5 = 7>5

3 + 1 = 4<5 â‡’ the ordered pairs is (3, 1)

3 + 2 = 5

3 + 3 = 6>5

3 + 4 = 7>5

3 + 5 = 8>5

4 + 1 = 5

4 + 2 = 6>5

4 + 3 = 7>5

4 + 4 = 8>5

4 + 5 = 9>5

5 + 1 = 6>5

5 + 2 = 7>5

5 + 3 = 8>5

5 + 4 = 9>5

5 + 5 = 10>5

Therefore, the set of ordered pairs satisfying x + y< 5 = {(1,1), (1,2), (1,3), (2, 1), (2,2), (3,1)}.

(iii) x + y > 8

So, we find the ordered pair such that x + y>8, where x and y belongs to set A = {1, 2, 3, 4, 5}

1 + 1 = 2<8

1 + 2 = 3<8

1 + 3 = 4<8

1 + 4 = 5<8

1 + 5 = 6<8

2 + 1 = 3<8

2 + 2 = 4<8

2 + 3 = 5<8

2 + 4 = 6<8

2 + 5 = 7<8

3 + 1 = 4<8

3 + 2 = 5<8

3 + 3 = 6<8

3 + 4 = 7<8

3 + 5 = 8

4 + 1 = <8

4 + 2 = 6<8

4 + 3 = 7<8

4 + 4 = 8

4 + 5 = 9>8, so one of the ordered pairs is (4, 5)

5 + 1 = 6<8

5 + 2 = 7<8

5 + 3 = 8

5 + 4 = 9>8, so one of the ordered pairs is (5, 4)

5 + 5 = 10>8, so one of the ordered pairs is (5, 5)

Therefore, the set of ordered pairs satisfying x + y > 8 = {(4, 5), (5, 4), (5,5)}.

6. Given R = {(x, y) : x, yÂ âˆˆÂ W, x2 + y2 = 25}. Find the domain and Range of R.

Solution:

According to the question,

R = {(x, y) : x, yÂ âˆˆW, x2Â + y2Â = 25}

R = {(0,5), (3,4), (4, 3), (5,0)}

The domain of R consists of all the first elements of all the ordered pairs of R.

Domain of R = {0, 3, 4, 5}

The range of R contains all the second elements of all the ordered pairs of R.

Range of R = {5, 4, 3, 0}

7. If R1Â = {(x, y) | y = 2x + 7, where xÂ âˆˆÂ RÂ and â€“ 5 â‰¤ x â‰¤ 5} is a relation. Then find the domain and Range of R1.

Solution:

According to the question,

R1Â = {(x, y) | y = 2x + 7, where xÂ âˆˆRÂ and â€“ 5 â‰¤ x â‰¤ 5} is a relation

The domain of R1Â consists of all the first elements of all the ordered pairs of R1, i.e., x,

It is also given â€“ 5 â‰¤ x â‰¤ 5.

Therefore,

Domain of R1Â = [â€“5, 5]

The range of R contains all the second elements of all the ordered pairs of R1, i.e., y

It is also given y = 2x + 7

Now xÂ âˆˆÂ [â€“5,5]

Multiply LHS and RHS by 2,

We get,

2x âˆˆ [â€“10, 10]

Adding LHS and RHS with 7,

We get,

2x + 7 âˆˆ [â€“3, 17]

Or, y âˆˆ [â€“3, 17]

So,

Range of R1Â = [â€“3, 17]

8. If R2Â = {(x, y) | x and y are integers and x2Â + y2Â = 64} is a relation. Then find R2.

Solution:

We have,

R2Â = {(x, y) |Â x and y are integers and x2Â + y2Â â€“ 64}

So, we get,

x2Â = 0 and y2Â = 64 or x2Â = 64 and y2Â = 0

x = 0 and y = Â±8 or x = Â±8 and y = 0

Therefore, R2Â = {(0, 8), (0, â€“8), (8,0), (â€“8,0)}

9. If R3Â = {(x, |x| ) |x is a real number} is a relation. Then find domain and range of R3.

Solution:

According to the question,

R3Â = {(x, |x|) |x is a real number} is a relation

Domain of R3Â consists of all the first elements of all the ordered pairs of R3, i.e., x,

It is also given that x is a real number,

So, Domain of R3Â = R

Range of R contains all the second elements of all the ordered pairs of R3, i.e., |x|

It is also given that x is a real number,

So, |x| = |R|

â‡’Â |x|â‰¥0,

i.e., |x| has all positive real numbers including 0

Hence,

Range of R3Â = [0, âˆž)

10. Is the given relation a function? Give reasons for your answer.

(i) h = {(4, 6), (3, 9), (â€“ 11, 6), (3, 11)}

(ii) f = {(x, x) | x is a real number}

(iii) g = n, (1/n) |n is a positive integer

(iv) s = {(n, n2) | n is a positive integer}

(v) t = {(x, 3) | x is a real number.

Solution:

(i) According to the question,

h = {(4, 6), (3, 9), (â€“ 11, 6), (3, 11)}

Therefore, element 3 has two images, namely, 9 and 11.

A relation is said to be function if every element of one set has one and only one image in other set.

Hence, h is not a function.

(ii) According to the question,

f = {(x, x) | x is a real number}

This means the relation f has elements which are real number.

Therefore, for every xÂ âˆˆÂ R there will be unique image.

A relation is said to be function if every element of one set has one and only one image in other set.

Hence, f is a function.

(iii) According to the question,

g = n, (1/n) |nÂ is a positive integer

Therefore, the element n is a positive integer and the corresponding 1/nÂ will be a unique and distinct number.

Therefore, every element in the domain has unique image.

A relation is said to be function if every element of one set has one and only one image in other set.

Hence, g is a function.

(iv) According to the question,

s = {(n, n2) | n is a positive integer}

Therefore, element n is a positive integer and the corresponding n2Â will be a unique and distinct number, as square of any positive integer is unique.

Therefore, every element in the domain has unique image.

A relation is said to be function if every element of one set has one and only one image in other set.

Hence, s is a function.

(v) According to the question,

t = {(x, 3) | x is a real number.

Therefore, the domain element x is a real number.

Also, range has one number i.e., 3 in it.

Therefore, for every element in the domain has the image 3, it is a constant function.

A relation is said to be function if every element of one set has one and only one image in other set.

Hence, t is a function.

11. If f and g are real functions defined by f (x) = x2Â + 7 and g (x) = 3x + 5, find each of the following
(a) f (3) + g (â€“ 5)

(b) f(Â½) Ã— g(14)

(c) f (â€“ 2) + g (â€“ 1)

(d) f (t) â€“ f (â€“ 2)

(e) (f(t) â€“ f(5))/ (t â€“ 5), if t â‰  5

Solution:

According to the question,

f and g are real functions such that f (x) = x2Â + 7 and g (x) = 3x + 5

(a) f (3) + g (â€“ 5)

f (x) = x2Â + 7

Substituting x = 3 in f(x), we get

f (3) = 32Â + 7 = 9 + 7 = 16 â€¦(i)

And,

g (x) = 3x + 5

Substituting x = â€“5 in g(x), we get

g (â€“5) = 3(â€“5) + 5 = â€“15 + 5 = â€“10â€¦â€¦â€¦â€¦(ii)

We get,

f (3) + g (â€“ 5) = 16â€“10 = 6

(b) f(Â½) Ã— g(14)

f (x) = x2Â + 7

Substituting x = Â½ in f(x), we get

f(Â½) = (Â½)2 + 7 = Â¼ + 7 = 29/4 â€¦(i)

And,

g (x) = 3x + 5

Substituting x = 14 in g(x), we get

g (14) = 3(14) + 5 = 42 + 5 = 47â€¦â€¦â€¦â€¦(ii)

Multiplying equation (i) and (ii),

We get,

f(Â½) Ã— g(14) = (29/4) Ã— 47 = 1363/4

(c) f (â€“ 2) + g (â€“ 1)

f (x) = x2Â + 7

Substituting x = â€“2 in f(x), we get

f (â€“2) = (â€“2)2Â + 7 = 4 + 7 = 11â€¦â€¦..(i)

And,

g (x) = 3x + 5

Substituting x = â€“1 in g(x), we get

g (â€“1) = 3(â€“1) + 5

= â€“3 + 5 = 2â€¦â€¦â€¦â€¦(ii)

We get,

f (â€“ 2) + g (â€“ 1) = 11 + 2 = 13

(d) f (t) â€“ f (â€“ 2)

f (x) = x2Â + 7

Substituting x = t in f(x), we get

f (t) = t2Â + 7â€¦â€¦..(i)

Considering the same function,

f (x) = x2Â + 7

Substituting x = â€“2 in f(x), we get

f (â€“2) = (â€“2)2Â + 7 = 4 + 7 = 11â€¦â€¦.(ii)

Subtracting equation (i) with (ii),

We get,

f (t) â€“ f (â€“ 2) = t2Â + 7 â€“ 11= t2 â€“ 4

(e)Â (f(t) â€“ f(5))/ (t â€“ 5), ifÂ t â‰  5

f (x) = x2Â + 7

Substituting x = t in f(x), we get

f (t) = t2Â + 7â€¦â€¦..(i)

Considering the same function,

f (x) = x2Â + 7

Substituting x = 5 in f(x), we get

f (5) = (5)2Â + 7 = 25 + 7 = 32â€¦â€¦..(ii)

From equation (i) and (ii), we get

12. Let f and g be real functions defined by f (x) = 2x + 1 and g (x) = 4x â€“ 7.
(a) For what real numbers x, f (x) = g (x)?

(b) For what real numbers x, f (x) < g (x)?

Solution:

According to the question,

f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x â€“ 7

(a) For what real numbers x, f (x) = g (x)

To satisfy the condition f(x) = g(x),

Should also satisfy,

2x + 1 = 4xâ€“7

â‡’Â 7 + 1 = 4xâ€“2x

â‡’Â 8 = 2x

Or, 2x = 8

â‡’Â x = 4

Hence, we get,

For x = 4, f (x) = g (x)

(b) For what real numbers x, f (x) < g (x)

To satisfy the condition f(x) < g(x),

Should also satisfy,

2x + 1 < 4xâ€“7

â‡’Â 7 + 1 < 4xâ€“2x

â‡’Â 8 < 2x

Or, 2x > 8

â‡’Â x > 4

Hence, we get,

For x > 4, f (x) > g (x)

13. If f and g are two real valued functions defined as f (x) = 2x + 1, g (x) = x2 + 1, then find.

(i) f + g (ii) f â€“ g (iii) fg (iv)f/g

Solution:

According to the question,

f and g be real valued functions defined as f (x) = 2x + 1, g (x) = x2Â + 1,

(i) f + g

â‡’Â f + g = f(x) + g(x)

= 2x + 1 + x2Â + 1

= x2Â + 2x + 2

(ii) f â€“ g

â‡’Â f â€“ g = f(x) â€“ g(x)

= 2x + 1 â€“ (x2Â + 1)

= 2x â€“ x2

(iii) fg

â‡’Â fg = f(x) g(x)

= (2x + 1)( x2Â + 1)

= 2x(x2Â ) + 2x(1) + 1(x2) + 1(1)

= 2x3Â + 2x + x2Â + 1

= 2x3Â + x2Â + 2x + 1

(iv) f/g

f/g = f(x)/g(x)

14. Express the following functions as set of ordered pairs and determine their range.

f: XÂ â†’Â R, f (x) = x3Â + 1, where X = {â€“1, 0, 3, 9, 7}

Solution:

According to the question,

A function f: XÂ â†’R, f (x) = x3Â + 1, where X = {â€“1, 0, 3, 9, 7}

Domain = f is a function such that the first elements of all the ordered pair belong to the set X = {â€“1, 0, 3, 9, 7}.

The second element of all the ordered pair are such that they satisfy the condition f (x) = x3Â + 1

When x = â€“ 1,

f (x) = x3Â + 1

f (â€“ 1) = (â€“ 1)3Â + 1 = â€“ 1 + 1 = 0 â‡’ ordered pair = (â€“1, 0)

When x = 0,

f (x) = x3Â + 1

f (0) = (0)3Â + 1 = 0 + 1 = 1â‡’ ordered pair = (0, 1)

When x = 3,

f (x) = x3Â + 1

f (3) = (3)3Â + 1 = 27 + 1 = 28â‡’ ordered pair = (3, 28)

When x = 9,

f (x) = x3Â + 1

f (9) = (9)3Â + 1 = 729 + 1 = 730â‡’ ordered pair = (9, 730)

When x = 7,

f (x) = x3Â + 1

f (7) = (7)3Â + 1 = 343 + 1 = 344â‡’ ordered pair = (7, 344)

Therefore, the given function as a set of ordered pairs is

f = {(â€“1, 0), (0, 1), (3, 28), (7, 344), (9, 730)}

And,

Range of f = {0, 1, 28, 730, 344}

15. Find the values of x for which the functions

f (x) = 3x2Â â€“ 1 and g (x) = 3 + x are equal

Solution:

According to the question,

f and g functions defined by f (x) = 3x2Â â€“ 1 and g (x) = 3 + x

For what real numbers x, f (x) = g (x)

To satisfy the condition f(x) = g(x),

Should also satisfy,

3x2Â â€“ 1 = 3 + x

â‡’Â 3x2Â â€“ x â€“ 3 â€“ 1 = 0

â‡’Â 3x2Â â€“ x â€“ 4 = 0

Splitting the middle term,

We get,

â‡’Â 3x2Â + 3x â€“ 4xâ€“4 = 0

â‡’Â 3x(x + 1) â€“ 4(x + 1) = 0

â‡’Â (3x â€“ 4)(x + 1) = 0

â‡’Â 3x â€“ 4 = 0 or x + 1 = 0

â‡’Â 3x = 4 or x = â€“1

â‡’ x = 4/3, â€“1

Hence, forÂ x = 4/3, â€“1, f (x) = g (x),

i.e., given functions are equal.

16. Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the relation, g (x) = Î±x + Î², then what values should be assigned to Î± and Î²?

Solution:

According to the question,

g = {(1, 1), (2, 3), (3, 5), (4, 7)}, and is described by relation g (x) = Î±x + Î²

Now, given the relation,

g = {(1, 1), (2, 3), (3, 5), (4, 7)}

g (x) = Î±x + Î²

For ordered pair (1,1), g (x) = Î±x + Î², becomes

g (1) = Î±(1) + Î² = 1

â‡’Â Î± + Î² = 1

â‡’Â Î± = 1 â€“ Î² â€¦(i)

Considering ordered pair (2, 3), g (x) = Î±x + Î², becomes

g (2) = Î±(2) + Î² = 3

â‡’Â 2Î± + Î² = 3

Substituting value of Î± from equation (i), we get

â‡’Â 2(2) + Î² = 3

â‡’Â Î² = 3 â€“ 4 = â€“ 1

Substituting value of Î² in equation (i), we get

Î± = 1â€“Î² = 1â€“(â€“1) = 2

Now, the given equation becomes,

i.e., g (x) = 2xâ€“1

17. Find the domain of each of the following functions given by

Solution:

(i)

According to the question,

We know the value of cos x lies between â€“1, 1,

â€“1 â‰¤ cos x â‰¤ 1

Multiplying by negative sign, we get

Or 1 â‰¥ â€“ cos x â‰¥ â€“1

2 â‰¥ 1â€“ cos x â‰¥ 0 â€¦(i)

Now,

1â€“ cosÂ x â‰  0

â‡’Â cos x â‰  1

Or, x â‰  2nÏ€Â âˆ€ n âˆˆ Z

Therefore, the domain of f = Râ€“{2nÏ€:nâˆˆZ}

(ii)

According to the question,

For real value of f,

x + |x| > 0

When x > 0,

x + |x| > 0â‡’Â x + x > 0 â‡’Â 2x > 0â‡’Â x > 0

When x < 0,

x + |x| > 0â‡’Â x â€“ x > 0 â‡’Â 2x > 0â‡’Â x > 0

So, x > 0, to satisfy the given equation.

Therefore, the domain of f = R+

(iii)

f(x) = x|x|

According to the question,

We know x and |x| are defined for all real values.

Therefore, the domain of f = R

(iv)

According to the question,

For real value of

x2â€“1â‰ 0

â‡’Â (xâ€“1)(x + 1)â‰ 0

â‡’Â xâ€“1â‰ 0 or x + 1â‰ 0

â‡’Â xâ‰ 1 or xâ‰ â€“1

Therefore, the domain of f = Râ€“{â€“1, 1}

(v)

According to the question,

For real value of

28 â€“ x â‰ 0

â‡’Â xâ‰  28

Therefore, the domain of f = Râ€“{28}

18. Find the range of the following functions given by

Solution:

(i)

â‡’Â y>0 and 2yâ€“3â‰¥0

â‡’Â y>0 and 2yâ‰¥3

â‡’Â y>0 andÂ y â‰¥ 3/2

Or f(x)>0 andÂ f(x) â‰¥ 3/2

f(x) âˆˆ ( â€“ âˆž, 0) âˆª [ 3/2 , âˆž)

Therefore, the range of f =Â ( â€“ âˆž, 0) âˆª [ 3/2 , âˆž)

(ii) f(x) = 1â€“|xâ€“2|

According to the question,

For real value of f,

|xâ€“2|â‰¥ 0

Or â€“|xâ€“2|â‰¤ 0

â‡’Â 1â€“|xâ€“2|â‰¤ 1

Or f(x)â‰¤1

â‡’Â f(x)âˆˆÂ (â€“âˆž, 1]

Therefore, the range of f = (â€“âˆž, 1]

(iii) f(x) = |xâ€“3|

According to the question,

We know |x| are defined for all real values.

And |xâ€“3| will always be greater than or equal to 0.

i.e., f(x) â‰¥ 0

Therefore, the range of f = [0, âˆž)

(iv) f (x) = 1 + 3 cos2x

According to the question,

We know the value of cos 2x lies between â€“1, 1, so

â€“1â‰¤ cos 2xâ‰¤ 1

Multiplying by 3, we get

â€“3â‰¤ 3cos 2xâ‰¤ 3

â€“2â‰¤ 1 + 3cos 2xâ‰¤ 4

Or, â€“2â‰¤ f(x)â‰¤ 4

Hence f(x)âˆˆÂ [â€“2, 4]

Therefore, the range of f = [â€“2, 4]

19. Redefine the functionÂ f(x) = |x â€“ 2| + |2 + x|, â€“ 3 â‰¤ x â‰¤ 3

Solution:

According to the question,

function f(x) = |xâ€“2| + |2 + x|, â€“3â‰¤ xâ‰¤ 3

We know that,

when x>0,

|x â€“ 2| is (xâ€“2), xâ‰¥2

|2 + x| is (2 + x), xâ‰¥â€“2

when x>0

|x â€“ 2| is â€“(xâ€“2), x<2

|2 + x| is â€“(2 + x), x<â€“2

Given that, f(x) = |xâ€“2| + |2 + x|, â€“3â‰¤ xâ‰¤ 3

It can be rewritten as,

20. IfÂ

, then show that:

Solution:

(i)

Hence proved

(ii)

Substituting x by â€“ 1/x, we get

Hence proved

21. Let f(x) = âˆšxÂ and g (x) = x be two functions defined in the domain R+âˆªÂ {0}. Find

(i) (f + g) (x)

(ii) (f â€“ g) (x)

(iii) (fg) (x)

(iv) (f/g) (x)

Solution:

(i)

(f + g)(x)

â‡’Â (f + g)(x) = f(x) + g(x)

â‡’Â f(x) + g(x) = âˆšx + x

(ii)

(f â€“ g)(x)

â‡’Â (f â€“ g)(x) = f(x) â€“ g(x)

â‡’Â f(x) â€“ g(x) = âˆšxâ€“x

(iii)

(fg)(x)

â‡’Â (fg)(x) = f(x) g(x)

â‡’Â (fg)(x) = (âˆšx)(x)

â‡’Â f(x)g(x)= xâˆšx

(iv)

(f/q)(x) = f(x)/g(x)