 Hi and welcome to the session. Let's discuss the following question which says that f is equal to the set of ordered pair x, comma x is square upon 1 plus x is square such that x belong to the set of real numbers via function from r to r determining the range of f. So first let us learn what is a function. Now a function from a set a to b is a specific type of relation for which every element x of set a 1 and only 1 image y in set b. That is f x is equal to y for all x belonging to set a and y belonging to set b and the range is equal to all the f x such that x belong to set y. So the definition is a key idea which we are going to use in this problem to solve it. Let us now start with the solution and here we are given function f is equal to set of all the ordered pair x, comma x square upon 1 plus x square such that x belongs to r we have to find the range of f such that r goes to r is a function fx is equal to x square upon 1 plus x square and let y is equal to fx which is the range of f then we have y is equal to x square upon 1 plus x square which further implies that x square is equal to y into 1 plus x square which we get on cross multiplying it or x square into 1 minus y is equal to y or x square is equal to y upon 1 minus y or x is equal to plus minus root over y upon 1 minus y and now since x belongs to r this implies y upon 1 minus y is greater than or equal to 0 which implies that 1 minus y is not equal to 0 or y is not equal to 1 also y upon 1 minus y is greater than or equal to 0 implies that y is greater than or equal to 0 and 1 minus y is greater than 0 or we can say that y is less than 1 therefore from these two we conclude that y is less than 1 and greater than or equal to 0 as we say that range of f is equal to the set having elements y this is equal to fx such that y is less than 1 and greater than or equal to 0 that is ranges any positive real number 0 is less than or equal to x is less than 1 so this is the range the solution of the problem hope you enjoyed the session take care and have a good day