Q. 14.1( 12 Votes )

# State with reasons whether the following functions have inverse:

(i) f : [1, 2, 3, 4] → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Answer :

(i) f : [1, 2, 3, 4] → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}

Recall that a function is invertible only when it is both one-one and onto.

Here, we have f(1) = 10 = f(2) = f(3) = f(4)

Hence, f is not one-one.

Thus, the function f does not have an inverse.

(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}

Recall that a function is invertible only when it is both one-one and onto.

Here, we have g(5) = 4 = g(7)

Hence, g is not one-one.

Thus, the function g does not have an inverse.

(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}

Recall that a function is invertible only when it is both one-one and onto.

Here, observe that distinct elements of the domain {2, 3, 4, 5} are mapped to distinct elements of the co-domain {7, 9, 11, 13}.

Hence, h is one-one.

Also, each element of the range {7, 9, 11, 13} is the image of some element of {2, 3, 4, 5}.

Hence, h is also onto.

Thus, the function h has an inverse.

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