3x^3+3x^2-18x/x^2+2x-3

Question

determine the domain, asymptotes,holes, and intercepts

Answer 1

(3x³+3x²-18x)/(x²+2x-3)

 

Factors to:

 

3x(x+3)(x-2)/(x+3)(x-1)

 

1) Domain is all x except x=-3 and x=1. 

 

Simplify by cancelling the (x+3) terms:

 

3x(x-2)/(x-1)

 

2) Asymptotes:

 

Vertical asymptote at x=1 (denominator goes to zero)

 

Since the numerator's degree is one greater than the denominator's. we have a slant asymptote.  Divide numerator by denominator to get asymptote's equation::

 

         _3x - 3__           Slant asymptote at y = 3x - 3

(x-1) )3x²-6x

         3x²- 3x

         ---------

              -3x

              -3x+3

              -------

                    -3

 

3) Hole at x = -3 (function not defined there - not in domain)

 

4) Intercepts:

 

x-intercepts where numerator = 0: (0,0) and (2,0)

 

y-intercepts when x= 0: (0,0)