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## Karnataka 1st PUC Maths Question Bank Chapter 2 Relations and Functions

Question 1.

Define ordered pair.

Answer :

Two numbers a and b listed in specific order and enclosed in parentheses form is called an ordered pair (a, b).

Keen Eye: Equality of two ordered pairs:

We have {a, b)-(c,d)⇔a-c and b – d.

Question 2.

Define a Cartesian product of two sets.

Answer :

Let A and B two non-empty sets. Then, the Cartesian product of A and B is the set denoted by Ax B, consisting of all ordered pairs (a, b) such that a e A and be B.

A x B= {(a, b): a ∈ A, b∈B}

Keen Eye:

- If n(A) = p and n(B) = q, then n (A x B) = pq and n (B x A) = pq
- If at least one of A and B is infinite then AxB is infinite and B x A is infinite,
- In general, A x B ≠ B x A
- A x A x A = {(a, b, c) : a, b, c ∈A}. Here (a, b, c) is called an ordered triplet.

Question 3.

If (x + 1, y – 2) = (3,1), find the values of x

Answer :

Given (x + 1, y – 2) = (3,1)

⇒ x+1=3 ∴x=2

y-2=1 ∴ y=3

Question 4.

If \( \left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)\)

Answer:

Given \( \left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)\)

Question 5.

If P={a,b,c}and Q={r},from P×Q and Q x P.Are these two products equal?

Answer:

PxQ = {(a,r),(b,r)(c,r)}

QxP = {(r, a), (r, b), (r, c)}

Clearly PxQ≠QxP

Question 6.

If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in

Answer :

Given n(A) = 3; n(B) = 3.

∴ n(AxB) = 3×3 = 9

Question 7.

If G=(7, 8} and H={5, 4, 2), find G x II and

Answer :

GxH = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}

HxG = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}

Question 8.

State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P={m, n} and Q = {n, m}, then P x Q = {(m, n), (n, m)}.

(ii) If A and B are non-empty sets, then A x B is a non-empty set of ordered pairs (at, y) such that x∈B and y∈A

(iii) If A = {1,2}, B = {3,4}, then A x {B∩φ ) = φ

Answer :

(i) Given statement is false:

Correct statement:

PxQ={(m, n), (m, m), (n, n), (n, m)}.

(ii) Given statement is false:

Correct statement:

AxB = {(x, y) :x∈A, y ∈B}. ‘

(iii) True statement,

Question 9.

If A x B = {(a, x), (a, y), (b, x), (b, y)}. Find A and

Answer :

A = {a, b} and B – {x, y}

Question 10.

If A x B = {(p, q), (p, r), (m, q), (m, r)}, find A and

Answer :

A = set of first elements = {p, m}

B = set of second elements = {q, r}

Question 11.

Let A = (1, 2}, B = [1, 2, 3, 4}, C = { 5, 6} and D = (5,6,7,8}. Verify that

(i) A x (B∩C) = (A x B)∩(A x C).

(ii) A x C is a subset

Answer :

(i) B∩C = { }

∴ Ax(B∩C)=φ ………….. (1)

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

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

∴ (A x B) ∩ (A x C) = φ ………………. (2)

From (1) and (2), we get

A x (B∩C) = (A x B)∩(A x C)

(ii) A x C = {(1, 5), (1,6), (2, 5), (2, 6)}

B x D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}.

Clearly every elements of A x C is an element of B x D.

A x C ⊂B x D.

Question 12.

Let A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}. Find

(i) A x (B ∩ C)

(ii) (A x B) ∩ (A x C)

(iii) A x (B∪C)

(iv) (A x B)∪(A x C)

Answer :

(i) B∩C={4}

A x (B∩C) = (1,4), (2, 4), (3,4)}

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

A x C = {(1, 4), (1, 5) (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

(A x B)∩(A x C)= {(1, 4), (2, 4), (3, 4)}

(iii) B ∪ C={3,4, 5, 6}

∴ Ax(B∪C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}

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

A x C = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}

(A x B)∪(A x C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6) (3, 3), (3, 4), (3, 5), (3,6)}.

Question 13.

Let A = {1, 2} and B = {3, 4}. Write A x B. How many subsets will A x B have? List them.

Answer :

Given A = {1, 2} and B = {3,4}

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

∴n (A x B) = 4

Number of subsets of A x B = 2^{4}=16

Subsets of A x B are: A x B, φ, {(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, {(1, 3), (1, 4)}, {(1, 3), (2, 3)}, {(1, 3), (2,4)}, {(1,4), (2, 3)}, {(1, 4), (2, 4)} {(2, 3), (2, 4)}, {(1, 3), (1, 4), (2, 3)}, {(1, 3), (1, 4), (2, 4)}, {(1,4), (2, 3), (2, 4)}, {(2, 3), (2,4), (1, 3)}.

Question 14.

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A x B, find A and B, where x, y, z are distinct elements.

Answer :

A = {x, y, z} and B = {1, 2}.

Question 15.

The Cartesian product A x A has 9 elements along which are found (-1, 0) and (0,1). Find the set A and the remaining elements Ax A.

Answer :

Given n(A x A) = 9 = 3^{2
}⇒n(A) = 3

But (-1, 0) and (0, 1) are in A x A

∴ A= {-1,0,1}.

Remaining elements of A x A: (-1, -1), (-1, 1),

(0,-1), (0,0), (1,-1), (1,0), (1,1).

Question 16.

If P = {1,2}, form the set

Answer :

P x P x P = {(1, 1, 1), (1, 1, 2), (1, 2, 1),

(1, 2,2), (2, 1,1), (2, 1, 2), (2, 2,1), (2, 2, 2)}

Question 17.

If A = {-1,1}, find A x A x A.

Answer :

A x A x A = {(-1, -1, -1), (-1, -I, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1, -1, 1), (1,1,-1), (1,1,1)}

Question 18.

If R is the set of all real numbers, what do the Cartesian products R x R and R x R x R represent?

Answer :

We have R x R = {(x, y) : x, y ∈ R } which represents the coordinates of all the points in two dimensional space and R x R x R = {(x, y, z) x,y,z ∈ R } which represents the coordinates of all the points in three-dimensional space.

Question 19.

Define a relation.

Answer :

A relation R from a non-empty set A to non empty set B is a subset of the Cartesian product A x B.

Question 20.

Define domain of a relation.

Answer :

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

Question 21.

Define range of a relation.

Answer :

The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the co-domain of the relation R.

Question 22.

Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x,y): y = x + 1}

(i) Depict this relation using an arrow diagram

(ii) Write down the domain, condomain and range of

Answer :

Given R = {(x, y): y = x + 1}

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

Domain = {1, 2, 3,4, 5,}; Co-domain = A

Range = {2, 3,4, 5, 6}.

Question 23.

Let A = {1, 2, 3, ………….14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, x,

y ∈ A}. Write down its domain, co-domain and range.

Answer :

Given R = {(x, y): 3x -y = 0, x, y ∈ A}

= {(1, 3), (2, 6), (3, 9), (4,12)}

Domain = {1, 2, 3,4}

Co-domain = A Range = {3, 6, 9,12}

Question 24.

Define a relation R on the set N of natural numbers by R – {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈ . N}. Depict this relationship using roster form. Write down the domain and the range.

Answer :

Given R = {(x, y): x, y ∈ N and y = x + 5, x < 4}

= {(1,6), (2,7), (3, 8)}

Domain = {1, 2, 3}

Range = {6, 7, 8}

Question 25.

A = {1, 2, 3, 5} and B = (4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y∈ B}. Write R in roster form.

Answer :

Given A = {1, 2, 3, 5} and B = {4, 6, 9}

R = {(x, y): the difference between x and y is odd; x∈ A,y∈B}

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

Question 26.

The figure shows a relationship between the sets P and Write this relation

(i) in set- builder-form

(ii) roster form. What is its domain and range?

Answer:

Given P={5,6,7} and Q={3,4,5}

(i) Set builder form

R= {(x,y):x-y = 2; x∈P,y ∈Q)

(ii) Roster form

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

Domain of R = P

Range of R = Q.

Question 27.

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a, b ∈A, b is exactly divisible by a},

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R

Answer :

Given A = { 1, 2, 3,4, 6}

R- {(a, b),a,b∈A,bis exactly divisible by a}

(i) Roster form:

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

(ii) Domain of R = {1, 2, 3,4, 6} = A

(iii) Range of R = {1, 2, 3,4, 6} = A

Question 28.

Determine the domain and range of the relation R .defined by R = {(x, x + 5): x e {0,1, 2,3,4,5}}.

Answer :

Given R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}.

Domain of R = {0, 1, 2, 3,4, 5}

Range of R = {5, 6, 7, 8, 9, 10}

Question 29.

Write the relation R = {(x, x^{3}) : x is a prime number less than 10} in roster form.

Answer :

Given R = {(x, x^{3}): x g {2, 3, 5, 7}}

= {(2, 8), (3, 27), (5, 125), (7, 343)}

Question 30.

Let A – {x, y, z} and B = (1, 2}. Find the number of relations from A to

Answer :

Given n(A) = 3 and n(B) = 2

∴ n (A x B) = 3 x 2 = 6

Number of relations from A to B = 2^{n (A x }^{B)} = 2^{6 }= 64

Question 31.

Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of

Answer :

Given: R = {(a, b): a, b ∈Z,a-b is an integer} Domain of R-Z Range of R = Z

Question 32.

Let R be a relation from Q to Q defined by R = {(a, b)\ a,b ∈Q and a – b ∈ Z}. Show that (a, a) g R, for all a ∈Q

(ii) (a, b) ∈ R implies that (b, a) ∈R

(iii) (a, b) ∈ R and (b, c) ∈R implies that (a, c) ∈ R.

Answer :

Given R = {{a, b): a,b∈Q and a-b ∈ Z)

(i) ∀ a ∈ Q, a – a = 0 ∈ Z ⇒ (a, a)∈R

(ii) Let (a, b) ∈ R⇒ a- b ∈ Z

b – a ∈ Z ⇒ (b, a) ∈ R

(iii) Let (a, b) and (b, c) g R ⇒ a – b ∈Z and

b – c ∈ Z

a- c = (a-b) + (b – c) ∈Z

∴ (a, c)∈ R

Question 33.

Let R be a relation from A to A defined by R = {(a, b): a,b ∈ N and a = b^{2}} Are the following true?

(i) (a, a) ∈ R for all a ∈ A

(ii) (a, b) ∈ R, implies (b, a) ∈ R

(iii) (a, b) ∈ R, (b, c) ∈ R, implies (a, c) ∈ R.

Justify your answer in each case.

Answer :

Given R= {(a, b): a,b∈N and a = b^{2}}

= {(1,1), (2,4), (3, 9), (4, 16),…}

(i) (a, a)∈ R for all a ∈ N is not true because (2, 2) ∉ R.

(ii) (a, b) ∈ R implies (b, a) ∈ R is not true, because (2,4) ∈ R but (4,2) ∉ R

(iii) (a, b) ∈ R, (b, c) ∈ R implies (a,c) ∈ R is not true because (2,4) and (4,16) ∈ R but (2,16) ∉ R.

Question 34.

Define a function.

Answer :

A relation/from a set A to a set B is said to be a function if every element of set A has one and only one image in set B, and

we write f : A → B

Question 35.

Define

(i) a real valued function

(ii) a real function

Answer :

A function which has either R or one of its subsets as its range is called a real valued function. A function f: A → B is said to be a real function if both A and B are subsets of R.

Question 36.

Let N be the set of natural numbers and the relation R be defined on N such that

R = {(x,y):y = 2x,x,y ∈ N} What is the domain, co-domain and range of R? Is this relation a function?

Answer :

Given R = {x, y): y = 2x; x, y ∈ N}

Domain of R = set of natural numbers Co-domain of

R = set of natural number

Range of R = set of even natural numbers

Clearly, every natural number is related to unique image, so this relation is a function.

Question 37.

Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?

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

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

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

Answer :

(i) Given R= {(2,1), (3,1), (4, 2)}

Here every element of domain is related to unique element of co-domain, so it is a function.

(ii) Given R = {(2, 2), (2,4), (3, 3), (4, 4)}

Since the element 2 has two images namely 2 and 4, so it is not a function.

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

Since every element of domain is related to unique

Question 38.

Let N be the set of natural numbers. Define a real valued function f: N → N by

f(x) = 2x + 1. Using this definition, complete the table given below:

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

y | F : (1) = | F : ( 2) = | F : (3) = | F : (4) = | F : (5) = | F : (6) = | F : (7) = |

Answer:

Given function is f(x) = 2x + 1.

X | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

y | F:(1)=3 | F:(2)=5 | F:(3)=7 | F:(4)=9 | F:(5)=11 | F:(6)=13 | F:(7)=15 |

Question 39.

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

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

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14,7)}

(iii) {(1,3), (1,5), (2,5)}.

Answer :

(i) Clearly, every element of domain is related to unique element of co-domain, so it is a function.

Domain = {2, 5, 8, 11, 14, 17}

Range = {1}

(ii) Clearly, every element of domain is related to unique element of co-domain, so it is a function.

Domain = {2,4, 6, 8,10,12,14}

Range = {1,2, 3,4, 5, 6,7}

(iii) 1 is related to two elements of co-domain, namely 3 and 5, so it is not a function.

Question 40.

Let A={1,2,3,4), B={1,5,9,11,15,16} and f = {(1,5),(2,9), (3,1), (4,5),(2, 11)). Are the

following true?

(i) f is a relation from A to B

(ii) f is a function from A to B. Justify your answer in each case.

Answer :

(i) Every element off is an element of A x B, so f is a relation.

(ii) ‘f’ is not a function the element 2 has two images.

Question 41.

Let f be the subset of Z x Z defined by f ={(ab,a +b):a,b ∈z) Is f a function from Z to Z? Justify your answer.

Answer :

Given f={(ab,a+b):a,b∈Z)

If a = 1 and b = 4 = ab = 4 and a+b=5

∴ (4,5) ∈ fIf a=2 and b=2⇒ab=4 and a+b=4

∴ (4,4) ∈ f∴ The element 4 has two images, so f is not a function.

Question 42.

The relation f is defined by

\( f(x)=\left\{\begin{array}{ll}{x^{2},} & {0 \leq x \leq 3} \\ {3 x,} & {3 \leq x \leq 10}\end{array}\right.\)

The relation g is defined by

\( g(x)=\left\{\begin{array}{ll}{x^{2},} & {0 \leq x \leq 2} \\ {3 x,} & {2 \leq x \leq 10}\end{array}\right.\) .

Show that/is a function and g is not a function.

Answer :

Since f(x) is unique for 0 ≤ x ≤ 10.

f(x) is a function. g(2) = 2^{2} =4 and g(2) = 3(2) = 6

∴ z has two images under g.

∴ g is not a function.

Question 43.

Let f= {(1,1),(2,3),(0,-1),(-1,-3)} be a linear function from Z into Z. Find f(x) ).

Answer :

Since/is a linear function.

∴f{x) = ax + b

But (0,-1) ∈ f . f(0) ∴ a(0) + b ⇒ -1 = b

Similarly, (1,1) ∈ f ∴ f(1) = a(1) +a(1)+b

⇒ 1=a+b ∴ a=2 ∴ f(x) = 2x -1

Question 44.

A function f is defined by f(x) = 2x-5. Write down the values of (i) f(0) (ii) f(17) (iii) f(-3).

Answer :

Given: f(x) = 2x-5

- f(0) = 2(0) – 5 = -5
- f(17) = 2(17)-5 = 29
- f(-3) = 2(-3) – 5 = -11

Question 45.

The function ‘t’ which maps temperature in degree Celsius into temperature in degree. Fahrenheit is defined by \( t(c)=\frac{9 c}{5}+32 \). Find

(i) t (0)

(ii) t (28)

(iii) t (-10)

(iv) The value of c, when t(c) – 212

Answer:

Question 46.

\(\text { If } f(x)=x^{2}, \text { find } \frac{f(1 \cdot 1)-f(1)}{1 \cdot 1-1}\)

Answer:

Question 47.

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) Given f(x) = 2-3x, x∈R,x>0

For x > 0,f(x) = 2 – 3x < 2

∴ Range of f= (-∞, 2)

(ii) Given f(x) = x^{2} + 2, x ∈ R

For x ∈ R, f(x) = x^{2} + 2 ≥ 2

Range of f = [2, ∞)

(iii) Given f(x) = x, x∈ R For r∈ E, f(x) = x∈R

Range of f = R

Question 48.

Let A = (9, 10, 11, 12, 13} and let f:A→N be defined by f(n) = the highest prime factor of n. Find the range of f.

Answer :

Given f(n) = the highest prime factor of n.

f(9) = the highest prime factor of 9 = 3

f(10) = the highest prime factor of 10 = 5

f(11) = the highest prime factor of 11 = 11

f(12) = the highest prime factor of 12 = 3

f(13) = the highest prime factor of 13 = 13

Range of f ={3,5,11,13}

Question 49.

\(\text { Let } f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbb{R}\right\}\) be a function from R to R .Determine the range of f.

Answer:

Question 50.

Find the domain of the function \(f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}\)

Answer:

Given \(f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}\)

f(x) is defined for all real numbers except x^{2} – 5x + 4 = 0

But x^{2} -5x + 4 = (x-4) (x-1)

Domain of f= R-{1,4}

Question 51.

Find the domain of the function

\( f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}\)

Answer:

\(f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12} \)

is not defined when

x^{2} – 8+12 = 0.

∴ (x-6)(x-2) = 0

∴ Domain of f = R-{2,6}

Question 52.

Find the domain and range of the following real functions:

(i) \( f(x)=-|x| \)

(ii) \(f(x)=\sqrt{9-x^{2}}\)

(iii) \(f(x)=\sqrt{x-1}\)

(iv) \(f(x)=|x-1|\)

Answer:

Remark: Operations of functions:

(i) Addition of two real functions:

Let f: X → R and g : X → R Then.

(f + g):X → R; (f + g)(x) = f(x) + g(x) for all x ∈ X

(ii) Difference of two real functions:

Let f : X → R and g : X → R Then.

(f – g): X → R; (f- g)(x) = f(x) – g(x) for all x ∈ X

(iii) Scalar multiplication of a function:

Let f: X → R and let ‘a’ he a scalar. Then,

(αf):X → R; (fg)(x) = f(x) for all x ∈ X

(iv) Multiplication of two real functions:

Let f: X → R and g : X → R .Then

(fg): X → R; (fg)(x)= f(x)g(x), for all x ∈ X

(v) Quotient of two real functions:

Let f: X → R and g : X → R for all x for which g(x) ≠ 0. Then

\( \left(\frac{f}{g}\right): X \rightarrow \mathbb{R} ;\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)} \)

Question 53.

Let f(x) = x^{2} and g(x) = 2x + 1 be two real functions. Find (f+g)(x),(f-g)(x)

\((f g)(x),\left(\frac{f}{g}\right)(x)\)

Answer:

Question 54.

Let \( f(x)=\sqrt{x} \text { and } g(x)=x \) be two functions defined over the set of non-negative real numbers. Find (f+ g)(x), (f – g)(x) \((f g)(x) \text { and }\left(\frac{f}{g}\right)(x)\)

Answer:

Question 55.

Let f,g: R→R be defined, respectively by f(x) = x + 1,g(x) = 2x-3- Find f + g,f-g and \( \frac{f}{g} \)

Answer:

Given: f(x) = x + 1,g(x) = 2x-3

(f + g)(x) = f (x) + g(x) = x + 1 + 2x-3 = 3x-2

(f – g)(x) = fix) – g(x) = x + 1-2x + 3 = -x + 4

\( \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{x+1}{2 x-3}, x \neq \frac{3}{2}\)

Question 56.

Define an identity function and draw its graph also find its domain and range.

Answer :

The function f: M → R; f(x) = x for a II x∈R is called an identity function on R.

Question 57.

Define a constant function and draw its graph also find its domain and range.

Answer :

Let c be a fixed real number. Then, the function f : R→R, f(x) = c for all x∈R is called the constant function.

f(x) is defined for all real number,

∴ Domain = M

Range = { c }

Question 58.

Define a polynomial function.

Answer :

A function f : M → R is said to be polynomial function if for each x in IR,

y = f (x) = a_{0} + a_{x}x + a_{2}x^{2} + ……………. + a_{n}x^{n
}where n is a non-negative integer and a_{0},a_{1},a_{2………………….}a_{n} ∈ R .

Question 59.

Draw the graph of the function f (x) = x^{2} and write its domain and range.

Answer:

Given function: f(x)= x^{2
}Domain = R

Range = set of non-negative reals.

Question 60.

Draw the graph of the function f: R → R defined by f(x) = x^{3} Find its domain and range.

Answer :

Let f : R → R: f(x) = x^{3}, ∀ ∈ R

Then, domain of f = R and range of f = R . we have

Question 61.

Define a rational function

Answer:

The functions of the type \(\frac{f(x)}{g(x)}\) where f(x) g(x) and g(x) are polynomial functions of x, defined in a domain, where g(x) ≠ 0

Question 62.

Let f : R – {0} → R defined by \( f(x)=\frac{1}{x}, \forall x \in \mathbb{R}-\{0\}\) . Find its domain and x range. Also, draw its graph.

Answer :

Given function is f : R – {0} → R defined by \( f(x)=\frac{1}{x} \)

∴ Domain = R-{0} and range =R-{0}

X | -4 | -2 | -1 | -0-5 | -0-25 | 0-25 | 0-5 | 1 | 2 |

f(x)=1/x | -0-25 | -0-5 | -1 | -2 | -4 | 4 | 2 | 1 | 0-5 |

Question 63.

Define a modulus function. Find its domain and range. Also, draw its graph.

Answer :

Let f: R → R defined by f(x) =1 x I, for each x ∈ R, is called modulus function.

\( \text { i.e., } f(x)=|x|=\left\{\begin{array}{ll}{x,} & {\text { if } x \geq 0} \\ {-x,} & {\text { if } x<0}\end{array}\right.\)

Domain = R

Range = set of non negative real numbers

Question 64.

Define Signum function. Draw its graph and find its domain and range.

Answer :

The function f: R → R defined by

\( f(x)=\left\{\begin{array}{lll}{1,} & {\text { if }} & {x>0} \\ {0,} & {\text { if }} & {x=0} \\ {-1,} & {\text { if }} & {x<0}\end{array}\right.\) is called signum function

We have

Domain = R

Range = {-1,0,1}

Question 65.

Define a greatest integer function. Draw its graph and find its domain and range.

Answer:

The function f : R → R define by f(x) = [x], x∈ R assumes the value of the greatest integer, less than or equal to x. Such a’ function is called the greatest integer function or step function. We have

[x] = -2 for – 2 ≤ x < -1

[x] = -1 for -1≤x<0

[x] = 0 for 0≤x≤1

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3.

Hence, domain of f = R and range = Z.

Question 66.

Define a linear function.

Answer :

The function f : R → R defined by f(x) = mx + c, x ∈ R is called linear function, where m and c are constant.

Question 67.

Let R be the set of real numbers. Define the real function f : R →R by f(x) = x + 10 and sketch the graph of this function.

Answer :

Given f(x) = x +10

We have

Question 68.

The function f is defined by

\( f(x)=\left\{\begin{array}{cl}{1-x,} & {x<0} \\ {1,} & {x=0} \\ {x+1,} & {x>0}\end{array}\right.\).

Draw the graph of f(x)

Answer:

We have