Question

Write a rational function f(x) such that f has vertical asymptotes at x = 3 and x = -1, no horizontal asymptote, and end behavior that can be modeled by y = 2x.

Answer 1

`f(x)=(2x^3-4x^2-6x)/(x^2-2x-3)`

Explanation:

Roots of a denominator in a rational function gives to us the vertical asymptotes. Hence we can take the denominator as

`(x-3)(x+1)=x^2-2x-3`

if we want that the end behavior as y=2x we can choose a polynomial whose factors cancel out with the denominator. Thus

`2x(x-3)(x+1)=2x^3-4x^2-6x`

Hence, the function is

`f(x)=(2x^3-4x^2-6x)/(x^2-2x-3)`

Hope this helps!!