Question

How do I find the asymptotes of `f(x)= (- 4x-x^2)/(2 +2x- x^3)`?

Answer 1

asymptotes:
`x=2`
`x=sqrt(2)`
`x=-sqrt(2)`

Explanation:

Start by simplifying the function:

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

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

`f(x)=(-x(x+4))/(-x(x^2-2)+2)`

`f(x)=(-x(x+4))/((-x+2)(x^2-2))`

`f(x)=(-x(x+4))/(-(x-2)(x^2-2))`

`f(x)=(color(red)cancelcolor(black)-x(x+4))/(color(red)cancelcolor(black)-(x-2)(x^2-2))`

`f(x)=(x(x+4))/((x-2)(x^2-2))`

Take each bracketed polynomial in the denominator, set it to cannot equal to `0`, and solve for `x`.

Finding the asymptotes

`1. x-2!=0`
`color(white)(ixxxx)x!=2`

`2. x^2-2!=0`
`color(white)(xxxx)x^2!=2`
`color(white)(xxxxx)x!=+-sqrt(2)`

The asymptotes are also the values which cannot be substituted into the equation such that the denominator would be `0`.

`:.`, the asymptotes are `x=2`, `x=sqrt(2)`, and `x=-sqrt(2)`.