 Hello and welcome to this session. In this session we will learn about testing for a function and this function. Now let us discuss testing for a function. Now we know that a function is a special type of a relation but all relations are not functions. The all functions are relations. So they check whether a given relation is a function of not by the family test. Examining that in effect that is the domain will use of examine whether members in the given relation is the relation is not a function. Now let us see some examples to clarify these tests. Now consider these three examples. Now in the first example I1 is a relation from A to B is representing the domain to I1. Now where you can see that it is not used that is it is not connected with any element of the set B. So the domain is representing the domain is the relation from A to B and the domain is fully used up whether all elements in the domain such B. But here for which is the element of set A that is the set B. Therefore R2 is not a function. For example here R2 is the relation from A to B is representing the domain and you can see that the domain is fully used up whether set B for set has only one image in the second set. That is the first members are different. Therefore this relation is satisfying. Both of them have a relation to be a function. Therefore on one test there is the vertical line test. On the test of vertical line it is not possible some examples to clarify this test. That in the first example circle is representing the graph of a relation. Now we have drawn a vertical line L the graph of this relation is possible to draw a vertical line that intersects the graph of the relation to one point. Therefore in the example 2 so we have drawn a vertical line M passing through the graph of this relation. You can see that this vertical is the graph of the relation point. Let this be point C of the relation at only one point. Theory behind this vertical line test. Now here the vertical line of the given relation of the given relation is not possible in the second case. As the vertical line let us discuss this function find by a single formula rather it is defined that is the function is defined relation function the value i such that f of x is equal to minus 1 when x is greater than 0. f of x is equal to 0 when x is equal to 0 and f of x is equal to 1 when x is greater than 0 minus 1 function it involves 3 equations. Now when x is less than 0 then f of x is equal to minus 1 0 and f of x is equal to 0 when x is greater than 0 then f of x is equal to 1. That means the function f assigns a number to every real number. Therefore the domain we are getting the different values of f of x values of f of x function. Therefore let's continue the elements minus 1 0 and 1. Now let us see one more example x is less than 0 and f of x is equal to 2x when x is greater than 0. Then we have to find f of minus 1 and f of 2. First of all we have to find f of minus is less than 0 and when x is less than 0 therefore this 1 will be equal to. Now replacing x with minus 1 here this will be minus 1 square minus 1 which is equal to 1 minus 1 which is equal to 0. So f of minus 1 is equal to 0 which is greater than greater than 0. So when x is equal to 2 which is equal to 4 that means the value of the function at minus 1 is equal to 0 and the value of the. Thus we can say that a piece function is not defined by a single formula rather it is defined in two or more parts. It changes depending on the value then it is called a piece function. So in this session we have learnt about piece function. We have enjoyed the session.