Question

For what values of x, if any, does `f(x) = 1/((5x+8)(x-6)` have vertical asymptotes?

Answer 1

`x=-8/5` and `x=6`

Explanation:

For a rational function such as the one presented, vertical asymptotes occur whenever the denominator of the function is equal to `0`. These are the "holes" in the function's domain since it's impossible to divide by `0`.

Now, we must find the times when

`(5x+8)(x-6)=0`

What we have here are two terms, `5x+8` and `x-6`, that are being multiplied and equalling `0`. This means that either one of the terms must be equal to `0`, but it doesn't matter which.

Thus, to find the times when the whole expression equals `0`, we can set either term equal to `0`.

`mathbf((1))`

`5x+8=0`

`5x=-8`

`x=-8/5`

There is a vertical asymptote at `x=-8/5`.

`mathbf((2))`

`x-6=0`

`x=6`

There is a vertical asymptote at `x=6`.

We can check a graph of the original function:

graph{1/((5x+8)(x-6)) [-7.75, 12.25, -4.915, 5.085]}