Pre-Calculus help

Question

Create a rational function that meets the given criteria:
Numerator is a quadratic function with a x-intercept at x=-2
denominator is a cubic function
There must be only one removable discontinuity at x=3
There must be 2 infinite discontinuities at x=-1 and x=2

Answer 1

Removable discontinuity at x=3.  The numerator and denominator has a factor of (x - 3). Using this fact, the numerator of the function can be

 

(x + 2)(x - 3) = x² - x - 6

 

 

The denominator is cubic.  This tells you that the denominator must have 3 linear factors.

 

An infinite discontinuity means you will have a vertical asymptote. 

 

Your denominator is then

 

(x - 3)(x + 1)(x - 2) = (x - 3)(x²- x - 2)

                              = x³ - x² - 2x - 3x² + 3x + 6

                              = x³ - 4x² + x + 6

 

Your rational function

 

f(x) = (x² - x - 6) / (x³ - 4x² + x + 6)