Question

For what values of x, if any, does `f(x) = 1/((x-1)(x-6))` have vertical asymptotes?

Answer 1

`x=1,6`

Explanation:

Vertical asymptotes occur where the domain is undefined. That is, the spots when an error arises whenever `x` is plugged in. For a rational function such as this one, such an error will occur whenever the denominator of a fraction is equal to `0`.

To find the spots where `f(x)` has vertical asymptotes, set the denominator `(x-1)(x-6)=0`.

`(x-1)(x-6)=0`

`x-1=0`
or
`x-6=0`

`x=1`
or
`x=6`

Check a graph. The vertical asymptotes should occur when `x=1,6`.

graph{1/((x-1)(x-6) [-7.51, 17.8, -6.34, 6.32]}