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

Algebra ->  Equations -> 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       Log On


   

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) About Me  (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 x%5E2%2B4 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 x%5E2 is either zero or a positive number no matter what x is.
  2. If you add x%5E2, 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 x%5E2-4 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:
x%5E2-4=0
We could factor this into:
%28x%2B2%29%2A%28x-2%29=0
In order for a product to be zero one of the factors must ne zero so:
x%2B2=0 or x-2=0
Solving each of these we get
x+=+-2 or x=2
These would be the vertical asymptotes for a (simplified) rational function with x%5E2-4 as the denominator (like q%28x%29=5%2F%28x%5E2-4%29).