Question

State with reason whether 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 1

(i) f : {1, 2, 3, 4} → {10} defined as:
f = {(1, 10), (2, 10), (3, 10), (4, 10)}
From the given definition of f, we can see that f is a many one function as: f(1) = f(2) = f(3) = f(4) = 10
∴f is not one-one.

Hence, function f does not have an inverse.

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(ii) g : {5, 6, 7, 8} → {1, 2, 3, 4} defined as:
g = {(5, 4), (6, 3), (7, 4), (8, 2)}
From the given definition of g, it is seen that g is a many one function as: g(5) = g(7) = 4.
∴ g is not one-one,

Hence, function g does not have an inverse.

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(iii) h : {2, 3, 4, 5} → {7, 9, 11, 13} defined as:
h = {(2, 7), (3, 9), (4, 11), (5, 13)}
It is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under h.
∴ Function h is one-one.
Also, h is onto since for every element y of the set {7, 9, 11, 13}, there exists an element x in the set {2, 3, 4, 5}such that h(x) = y.

Thus, h is a one-one and onto function. Hence, h has an inverse.