Question

How do you write a rational function if the asymptotes of the graph are at `x=3` and `y=1`?

Answer 1

`f(x) = x/(x-3)`

Explanation:

Rational functions have vertical asymptotes where the denominator is zero and the numerator is non-zero (or has a zero of lower multiplicity). In our example we want a factor `(x-3)` in the denominator.

Rational functions have non-zero horizontal asymptotes if the numerator and denominator are of equal degree. The `y` value of the asymptote is given by the ratio of the coefficients of the leading terms of the numerator and denominator. In our example we want the leading coefficient of the numerator to be equal to the leading coefficient of the denominator, in order to get an asymptote `y=1`

So the simplest rational function with the desired behaviour would be:

`f(x) = x/(x-3)`

graph{x/(x-3) [-8.5, 11.5, -4.64, 5.36]}