Question

Can someone explain to me how holes work?
I'm in Pre-Cal and I just don't get it.

Answer 1

Well, the difference between holes and asymptotes is basically whether or not things cancel in a rational function. For example, let's say we have:

\[\frac{ x ^{2}+2x+1 }{ x ^{2}-2x-3 }\]Now we want to check and see if our equation has any values we have to exclude from the domain. So what we do is factor the bottomand solve for x. If we factor the bottom we get:

\[\frac{ x ^{2}+2x+1 }{ (x-3)(x+1) }\]

So this means we have to exclude 3 and -1 from the domain. But these two x values could be asymptotes or holes. Here is how we know: Let's factor the top
\[\frac{ (x+1)(x+1) }{ (x-3)(x+1) }\]
Notice how we would have an (x+1) factor cancel? If one of your factoes cancels, then the corresponding x-value is a hole and not an asymptote. Since the (x-3) does not cancel, x = 3 is an asymptote while x = -1 is a hole. So basically youre lookign to see if something cancels.