 Hello and welcome to the session. Let us study the following problem today. Show that the function f such that r star to r star, defined by fx is equal to 1 by x is 1, 1 and on to where r star is a set of all non-zero real numbers. Is the result true if the domain r star is replaced by n with co-domain being same as r star? Now let us write the key idea first. Prove 1, 1 are injective. We consider the definition a function f such that from x to y is said to be 1, 1. If for every x1, x2 belongs to x, f of x1 is equal to f of x2 which implies x1 is equal to x2. Now to prove on to or surjective, we consider the definition a function f such that from x to y is said to be on to if for every y belongs to y there exist an element x belongs to x such that fx is equal to y. Now let us write the solution. First, injectivity. f is a mapping from r star to r star. Let x, y belongs to r star where r star is a set of all non-zero real numbers. Now let x, y belongs to r star such that fx is equal to f of y then we have 1 by x is equal to 1 by y because fx is equal to 1 by x given to us which implies x is equal to y and thus f such that r star goes to r star is 1, 1. Now let us see for subjectivity that is on to let y be an arbitrary element r star that is co-domain such that fx is equal to y then fx is equal to y which implies 1 by x is equal to y because fx is equal to 1 by x given to us which implies x is equal to 1 by y and clearly 1 by y belongs to r star of domain for all y belongs to r star of co-domain thus we have proved for each y belongs to r star of co-domain there exist x is equal to 1 by y belongs to r star of domain such that fx is equal to 1 by x which is equal to y thus f is on to now let us consider f such that from n to r star given by fx is equal to 1 by x let us name it as 1 now let us prove for 1, 1 for every x y belongs to n we have fx is equal to f of y which implies 1 by x is equal to 1 by y by 1 which implies x is equal to y thus f from n to r star is 1, 1. Now let us prove for on to since co-domain r star is a set of all nonzero real numbers which contains elements like f5, 3 by 7 etc which do not have their pre-image in the domain and of natural numbers f such that from n to r star is not on to. I hope you understood this problem bye and have a nice day.