Question

How do you find the asymptote of a function

Answer 1

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:
$f(x) = \frac{x + 2}{x - 2}$
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;
$x - 2$
$x = 2 \: will \: be \: the \: horizontal \: asymptote$
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;
$f(x) = \frac{x + 2}{x - 2}$
We now write y instead of f(x).
$y = \frac{x + 2}{x - 2}$
Now, interchange the variables.
$x = \frac{y + 2}{y - 2}$
Now,make y the subject in order to get the inverse of the function.
$x(y - 2) = y + 2$
$xy - 2x = y + 2$
$xy - y = 2x + 2$
$y(x - 1) = 2x + 2$
$y = \frac{2x + 2}{x - 1}$
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.
$x - 1$
$x = 1$
Now, replace x with letter y.
$y = 1 \: is \: the \: vertical \: asymptote$
Therefore, the asymptotes have been obtained.
I hope you understand the concept.
Thank you;
kaloliavivek