SOLUTION: Give the equation of the vertical asymptote(s) of the rational function. F(x) = (x-1)/(x^2+4) A) x = 1, x = -1 B) x = 4 C) x = -4 D) None

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Question 201553: Give the equation of the vertical asymptote(s) of the rational function.
F(x) = (x-1)/(x^2+4)
A) x = 1, x = -1
B) x = 4
C) x = -4
D) None
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Vertical asymptotes of a rational function, if any, will be for values of x that make the denominator zero.

For your function we will try to find what values of x make negative. But if one understands how squaring works we can see that this particular denominator can never be zero because:
  1. When you square zero you get zero. When you square any other Real number you get some positive number. So is either zero or a positive number no matter what x is.
  2. If you add , which is zero or positive, to 4 you will always get a positive number, never a zero!
Since the denominator can never be zero, there are no vertical asymptotes for this function.

If you denominator had been instead, we could not say that the denominator could never be zero because subtracting 4 from a zero or positive number could result in a zero. So we would have to solve the equation:

We could factor this into:

In order for a product to be zero one of the factors must ne zero so:
or
Solving each of these we get
or
These would be the vertical asymptotes for a (simplified) rational function with as the denominator (like ).
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