There are two types of aymptotes; horizontal and vertical asymptote. To find asymptotes of a function, you first must know the domain and range of the given function. Let's say the function is: [tex]f(x) = \frac{x + 2}{x - 2} [/tex] Horizontal Asymptote: To find this asymptote, we have to look at the domain (or the denominator). The domain will be: x such that x is not = to 2. This is because the denominator is x-2, and if you substitute x with 2 it will be 2-2=0. But, anything divided by 0 is undefined. To find the asymptote,you take the denominator itself; [tex]x - 2[/tex] [tex]x = 2 \: will \: be \: the \: horizontal \: asymptote[/tex] Vertical Asymptote: To find this asymptote, we have to look at the range of the function by finding the inverse of the given function. From the given function; [tex]f(x) = \frac{x + 2}{x - 2} [/tex] We now write y instead of f(x). [tex]y = \frac{x + 2}{x - 2} [/tex] Now, interchange the variables. [tex]x = \frac{y + 2}{y - 2} [/tex] Now,make y the subject in order to get the inverse of the function. [tex]x(y - 2) = y + 2[/tex] [tex]xy - 2x = y + 2[/tex] [tex]xy - y = 2x + 2[/tex] [tex]y(x - 1) = 2x + 2[/tex] [tex]y = \frac{2x + 2}{x - 1} [/tex] The range will be x is not equal to 1, since the result will be 0, and, Therefore, the asymptote will be obtained by looking at the denominator of the inverse of the function; i.e. [tex]x - 1[/tex] [tex]x = 1 [/tex] Now, replace x with letter y. [tex]y = 1 \: is \: the \: vertical \: asymptote[/tex] Therefore, the asymptotes have been obtained. I hope you understand the concept. Thank you; kaloliavivek