Rational Function

Question

Given the rational function??(??) =(t³+t²-6t)/(t²-4) , find the domain, intercepts, holes, and asymptotes. Sketch an accurate graph of the ??rational function?

Answer 1

Factor the numerator and denominator.

 

Numerator:

 

t³ + t² - 6t = t(t² + t - 6) = t(t - 2)(t + 3)

 

Denominator:

 

t² - 4 = (t - 2)(t + 2)

 

Since (t - 2) occurs in both the numerator and denominator the same number of times, there is a hole at t = 2.

 

Since (t + 2) is in the denominator only, there is a vertical asymptote at t = -2.

 

Since t and (t + 3) are in the numerator only, the intercepts are at t = 0 and t = -3.