State with reason 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)}

Solution:

The inverse of a function f(x) is a function g(x) such that if f maps an element ′a′ to an element ′b′, g maps ′b′ to ′a′.

(i).

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

f is a many one function as

f (1) = f (2) = f (3) = f (4) = 10

Therefore,

f is not one-one.

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)}

g is a many one function as

g (5) = g (7) = 4

Therefore

g is not one-one.

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)}

All distinct elements of the set {2, 3, 4, 5} have distinct images under h .

Therefore,

h is one-one.

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.

h is a one-one and onto function.

Function h has an inverse

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.3 Question 5

State with reason 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)}

Summary:

f is not one-one. Function f does not have an inverse. g is not one-one. Function g does not have an inverse.  h is a one-one and onto function. Function h has an inverse