Find the coordinates of the hole:

Question

Find the coordinates of the hole of 

f(x) = x+1/x^2-4x-5

 

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Answer 1

f(x) = (x+1)/(x²-4x-5)

 

f(x) = (x+1)/[(x-5)(x+1)]

 

f(x) is undefined at x = 5 and x = -1 (since they would cause a division by 0).  However, we can cancel the (x+1) in the numerator with the (x+1) in the denominator:

 

f(x) = 1/(x-5)

 

Now we have a vertical asymptote at x = 5.  However f(x) is still undefined at x = -1, only now we have a "hole" (undefined point) there rather than a vertical asymptote.  The coordinates of the hole are (-1,f(-1)).  Plug x = -1 into the simplified f(x) (= 1/(x-5)) to find the value of f(-1).