Question

Which of the following are true for the rational function f(x)=x^2-2x-3/x

A) the horizontal asymptote is y=0
B) the function shown is f(x)=(x-3)(x+1)/x
C) the vertical asymptote is y=3
D) The zeros of the function are 1 and 3

Answer 1

B) The function shown is
`f(x)=((x-3)(x+1))/(x)`

Explanation:

The given function is

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

This is an improper rational function because the degree of the numerator is greater than the degree of the denominator. This means the function has a slant asymptote and cannot have a horizontal asymptote.

The vetical asymptote is
`x=0`.

We factor the numerator to get;

`f(x)=(x^2-3x+x-3)/(x)`

`f(x)=(x(x-3)+1(x-3))/(x)`

The function shown is the same as
`f(x)=((x-3)(x+1))/(x)`

The zeros of this function are
`x=-1,x=3`

The correct answer is B