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

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

Question 1.
Define ordered pair.
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.
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
Given (x + 1, y – 2) = (3,1)
⇒ x+1=3  ∴x=2
y-2=1  ∴ y=3

Question 4.
If $$\left(\frac{x}{3}+1, y-\frac{2}{3}\right)=\left(\frac{5}{3}, \frac{1}{3}\right)$$
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?
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
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
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
(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
A = {a, b} and B – {x, y}

Question 10.
If A x B = {(p, q), (p, r), (m, q), (m, r)}, find A and
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
(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)
(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.
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 = 24=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.
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.
Given n(A x A) = 9 = 32
⇒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
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.
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?
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.
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.
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.
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
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.
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.
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.
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?

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
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}}.
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, x3) : x is a prime number less than 10} in roster form.
Given R = {(x, x3): 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
Given n(A) = 3 and n(B) = 2
∴ n (A x B) = 3 x 2 = 6
Number of relations from A to B = 2n (A x B) = 26 = 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
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.
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 = b2} 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.
Given R= {(a, b): a,b∈N and a = b2}
= {(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.
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
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?
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)}
(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) =

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)}.
(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.
(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.
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.
Since f(x) is unique for 0 ≤ x ≤ 10.
f(x) is a function. g(2) = 22 =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) ).
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).
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

Question 46.
$$\text { If } f(x)=x^{2}, \text { find } \frac{f(1 \cdot 1)-f(1)}{1 \cdot 1-1}$$

Question 47.
Find the range of each of the following functions:
(i) f{x) = 2-3x, x∈R,x>0
(ii) f(x) = x2 + 2, x is a real number
(iii) f(x) = x, x is a real number.
(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) = x2 + 2, x ∈ R
For x ∈ R, f(x) = x2 + 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.
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.

Question 50.
Find the domain of the function $$f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$$
Given $$f(x)=\frac{x^{2}+3 x+5}{x^{2}-5 x+4}$$
f(x) is defined for all real numbers except x2 – 5x + 4 = 0
But x2 -5x + 4 = (x-4) (x-1)
Domain of f= R-{1,4}

Question 51.
Find the domain of the function
$$f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$$
$$f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$$
is not defined when
x2 – 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) $$f(x)=|x-1|$$

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) = x2 and g(x) = 2x + 1 be two real functions. Find (f+g)(x),(f-g)(x)
$$(f g)(x),\left(\frac{f}{g}\right)(x)$$

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) $$(f g)(x) \text { and }\left(\frac{f}{g}\right)(x)$$

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 $$\frac{f}{g}$$
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.
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.
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.
A function f : M → R is said to be polynomial function if for each x in IR,
y = f (x) = a0 + axx + a2x2 + ……………. + anxn
where n is a non-negative integer and a0,a1,a2………………….an ∈ R .

Question 59.
Draw the graph of the function f (x) = x2 and write its domain and range.
Given function: f(x)= x2

Domain = R
Range = set of non-negative reals.

Question 60.
Draw the graph of the function f: R → R defined by f(x) = x3 Find its domain and range.
Let f : R → R: f(x) = x3, ∀ ∈ R
Then, domain of f = R and range of f = R . we have

Question 61.
Define a rational function
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.
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.
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.
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.

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