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

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) = f (2) = f (3)

= f (4) = 10

⇒ f is not one-one

∴ f does not have an inverse

(ii) g(5) = 4 g(6) = 3 g(7)

= 4 g(8) = 2 g(5)

= g(7) = 4

∴ g is not one-one, hence g does not have inverse

(iii) h(2) = 7, h(3) = 9,

h(4) = 11, h(5)=13,

∴ h is one-one Also range of

h = {7,9,11,13}

= co-domain

⇒ h is onto

Hence h has an inverse h-1

= {(7,2), (9,3), (11,4), (13, 5)}.