Question

Show that the function f:N→N, given by f(1)=f(2)=1 and f(x)=x−1 for every x>2, is onto but not one-one.

Answer 1

The function f:N → N is onto but not one-one.

Step-by-step explanation:

To show that the function f:N → N, given by f(1)=f(2)=1 and f(x)=x-1 for every x>2, is onto but not one-one, we need to prove two properties. To show that f is onto, we need to demonstrate that every element of the co-domain (N) has at least one pre-image in the domain (N). Since f(1)=f(2)=1, both 1 and 2 in the co-domain are covered.

For every x>2, f(x)=x-1, so all other elements in the co-domain are also covered. Therefore, f is onto. To show that f is not one-one, we need to find two distinct elements in the domain that have the same image in the co-domain. Since f(1)=f(2)=1, we have found such elements, which means f is not one-one.