Determine which statement is accurate for the given function: ​​\(f(x) = {x^2-9 \over x+3}\)


a) The graph will have a hole.


b) The graph will have a vertical asymptote.


c) The graph will have a horizontal asymptote.


d) The graph will have a slant asymptote.



I found a) and d) to be correct. I can find both in the equation but am I only supposed to look for one? I'm confused.


For slant asymptotes, I got  \(y = x-3\)


For the Hole, I got  \(x=-3\)

 Apr 15, 2021

(x-3)(x+3)  / (x+3)           it will have a hole when the denominator = 0   at x = -3 


it reduces to  x-3  <====   a line   with a hole at   x = -3

 Apr 15, 2021

So the answer is a) ? But it also has a slant asymptote, doesn't it?

SmartMathMan  Apr 15, 2021

I don't think so....since part of the numerator cancels with the denominator it is just a line with a hole...

   if the numerator was something like   x^2-8    it would have a slant (oblique) asymtope   

ElectricPavlov  Apr 15, 2021

Oh. I think I understand. Thank you both!

SmartMathMan  Apr 15, 2021

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