Question 5

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

(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.

(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.

(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.

-->