 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that identify the horizontal and vertical asymptotes of the function f of x is equal to x plus 4 whole upon minus of 2x minus 6. Now let us start with the solution of the given question. We are given f of x is equal to x plus 4 whole upon minus 2x minus 6 and we have to find its horizontal and vertical asymptotes. First we see that the given rational function should be in simplified form that is if they are any common factors in the numerator and denominator then they should be cancelled. Here f of x is equal to x plus 4 whole upon minus 2x minus 6 which is already in simplified form. There are no common factors in numerator and denominator. So first we find its vertical asymptotes. The given function is of the form p of x upon q of x. For vertical asymptote we equate the denominator to 0 and obtain the value of x. So here q of x is equal to minus 2x minus 6 and we put minus 2x minus 6 is equal to 0 which implies that minus 2x is equal to 6 that is x is equal to 6 upon minus 2 which is equal to minus 3. So x is equal to minus 3. So we say that there will be a vertical asymptote at x is equal to minus 3. Now we have to find the horizontal asymptote. We know that in our rational function f of x is equal to p of x upon q of x. We determine the horizontal asymptotes by looking at the degree of both numerator and denominator of the given function. If degree of numerator is less than degree of denominator then horizontal asymptote is always the line y is equal to 0. If degree of numerator is equal to degree of denominator then the horizontal asymptote is obtained by dividing the leading coefficient of the numerator by the leading coefficient of the denominator and if degree of numerator is greater than degree of denominator then we get a slant line as the asymptote and it is called slant asymptote. Now in this function f of x is equal to x plus 4 whole upon minus 2x minus 6. Here degree of numerator is equal to degree of denominator and it is equal to 1 and we have just seen that if degree of numerator is equal to degree of denominator then the horizontal asymptote is obtained by dividing the leading coefficient of the numerator by the leading coefficient of the denominator. In numerator 1 is the leading coefficient and in denominator minus 2 is the leading coefficient so the horizontal asymptote is given by y is equal to 1 upon minus 2 which can be written as minus 1 upon 2 that is y is equal to minus 1 by 2 is the horizontal asymptote and x is equal to minus 3 is the vertical asymptote. Now let us see its graph. Now here in this graph we can see that at x is equal to minus 3 we have drawn the vertical asymptote and at y is equal to minus 1 by 2 we have drawn the horizontal asymptote. Now see the branches of the curve are becoming closer to these asymptotes but never intersect them. This is the required answer. This completes our session. Hope you enjoyed this session.