find the vertical asymptotes of the graph

Question

find the vertical asymptotes of the graph of 

 

f(x)= (x+1)/(x²+3x-10)

please show step by step 

Answer 1

The function f(x) is rational function.  To find vertical asymptotes of a rational function, one looks at the polynomial in the denominator.  In this case that denominator polynomial  is g(x) = x²+3x -10.  Vertical asymptotes (if any) will be located at the zeros of g(x).  (That is the values of x that satisfy g(x) =0).  Finding  the zeros of this particular g(x) is easy because it factors:

  

   g(x) = x² + 3x -10 = (x+5) (x-2)

 

So the zeros are x = -5  and x = 2.  

 

Next, one must check for a factor of (x +5)  or  (x-2) in the numerator polynomial.  In this case there is not. 

 

Thus there are two vertical asymptotes -  located at x = -5   and  x = 2.   The asymptotes are vertical lines. The equations of these two lines are:

 

     x = -5          and   x = 2

 

If it had not been possible to factor g(x), one would have used the quadratic formula or a calculator method to find the zeros (if any) of g(x).   Remember , if a zero is found for the denominator polynomial, one must check to see that the numerator does not also have a zero at the same value of x. If it does, then there is no corresponding vertical asymptote.

 

 

Answer 2

f(x) = (x+1)/(x² +3x -10)

 

Denominator can be factored into (x + 5)(x - 2)

 

The vertical asymptotes are those values of x for which the denominator becomes 0.

 

In this case, both x = -5 and x = 2 would cause the denominator to be 0.

 

The vertical asymptotes are therefore the lines x = -5 and x = 2