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)}

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Solution

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

Since 2, 5, 8, 11, 14, and 17 are the elements of the domain of the given relation having their unique images, this relation is a function.

Here, domain = {2, 5, 8, 11, 14, 17} and range = {1}

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

Since 2, 4, 6, 8, 10, 12, and 14 are the elements of the domain of the given relation having their unique images, this relation is a function.

Here, domain = {2, 4, 6, 8, 10, 12, 14} and range = {1, 2, 3, 4, 5, 6, 7}

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

Since the same first element i.e., 1 corresponds to two different images i.e., 3 and 5, this relation is not a function.

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Similar questions

Q. 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)}

Q. If X= { 1, 2, 3, 4, 5 }, Y= { 1, 3, 5, 7, 9 } determine which of the following relations from X to Y are functions? Give reason for your answer. If it is a function, state its type.
(i)

R_{1}

={(x, y)| y=x+2, x

ϵ

X, y

ϵ

Y}
(ii)

R_{2}

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

R_{3}

= { (1, 1), (1, 3), (3, 5), (3, 7), (5, 7) }
(iv)

R_{4}

= { (1, 3), (2, 5), (4, 7), (5, 9), (3, 1) }

Q. Let A = { 10, 11, 12, 13, 14 }; B = { 0, 1, 2, 3, 5 } and

f i : A \to B, i = 1, 2, 3

. State the type of function for the following (give reason):
(i)

f_{1}

= { (10, 1), (11, 2), (12, 3), (13, 5), (14, 3) }
(ii)

f_{2}

= { (10, 1), (11, 1), (12, 1), (13, 1), (14, 1) }
(iii)

f_{3}

= { (10, 0), (11, 1), (12, 2), (13, 3), (14, 5) }

Prove that:
(i) $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
(ii) $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$

Which of the following relations are functions from A = { 1, 4, 9, 16 } to B = { -1, 2, -3, -4, 5, 6 }. In case of a function, write down its range.

f_{4}

= { (1, 2), (4, 5), (9, -4), (16, 5) }

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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)}

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)}