 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says the relation f is defined by fx is equal to x square if x is greater than or equal to 0 and less than 3 and 3x when x is greater than or equal to 3 and less than or equal to 10 and the relation g is defined by gx is equal to x square if x is greater than or equal to 0 and less than or equal to 2, 3x if x is less than or equal to 10 and greater than or equal to 2. Show that x is a function and g is not a function. First let us learn that if a and b are any two sets then a function f which is defined from a to b is a specific type of relation for which every element x of set a has one and only one image y in set b that is if x belong to set a then f of x is equal to y where y belongs to set b and the element x of set a has only one and one image in set b. So this definition we will be using in this problem to solve it so this is a key idea. First let us show that f is a function x is defined by x square such that 0 is less than or equal to x is less than or equal to 3 and 3x such that 3 is less than or equal to x is less than or equal to 10. Now let fx is equal to y to show f is a function we will show that for every x there is a unique y. The first let x is equal to 0 then y which is equal to x square on substituting x as 0 get equal to 0 then let x is equal to 1 and y which is x square we have 1 square gives y is equal to 1 and let x is equal to 2 then y which is x square will be equal to 4 and when x is equal to 3 y has 2 values first is y is equal to x square which is equal to 9 and the second one is y is equal to 3x and on substituting x as 3 y becomes equal to 9 so for x is equal to 3 y has a unique value which is 9 and proceeding further let x is equal to 4 then y will be equal to 3 into 4 that is 12 and similarly like this taking x is equal to 10 y is equal to 3 into 10 that is 30 so we see that for every x there is a unique y hence include that for each x that x is less than or equal to 10 and greater than or equal to 0 is a unique y such that ordered pair x y belong to the function f this implies that the relation f no to ordered pair have the same first element and therefore f is a function now let us show that g is not a function now gx is defined by x square such that 0 is less than or equal to x is less than or equal to 2 and 3x such that 2 is less than or equal to x is less than or equal to 10 now let x is equal to y and to show that g is not a function we will have to show that for some x there are 2 different values of y so let us start with the x taking value 0 that is when x is equal to 0 then y which is equal to x square will be 0 when x is 1 y is 1 when x is equal to 2 y have 2 values first is y is equal to x square and next is when x is equal to 2 y is again 3x and on substituting x is 2 we get 2 square is 4 and on substituting 2 here we get 2 into 3 is 6 and we see that for x is equal to 2 we have 2 different values of y and this shows that for value of x we have 2 different values of y so this implies the order that 2 and 4 and 2 and 6 belong to g for x is less than or equal to 10 greater than or equal to 0 so this relation is not a function since they have the first element same which is 2 and hence g is not a function so this completes the solution hope you enjoyed it take care and bye for now