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

$\textbf{condition of inverse function:}\\\textbf{function should be one - one and onto}$

(i) f : {1,2,3,4} → {10} with f = {(1,10),(2,10),(3,10),(4,10)}
here we can see that f(1) = f(2) = f(3) = f(4) = 10
but 1 ≠ 2 ≠ 3 ≠ 4
therefore , f is not one- one function.
so, f is not inversible.

(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
here we can see that g(5) = g(7) = 4
but 5 ≠ 7
therefore, g is not one - one function.
so, g is not inversible.

(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
We can see that all distinct elements of the set (2, 3, 4, 5} have distinct images under h.
⇒ h is one–one.
Also, h is onto since for every element of the set {7, 9, 11, 13},
there exists an element x in the set {2, 3, 4, 5} such that h(x) = y.
Therefore, h is a one–one and onto function.
Therefore, h has an inverse.