Activity 5.5.2.
For each of the following rational functions, state the function’s domain and determine the locations of all zeros, vertical asymptotes, and holes. Provide clear justification for your work by discussing the zeros of the numerator and denominator, as well as a table of values of the function near any point where you believe the function has a hole. In addition, state the value of the horizontal asymptote of the function or explain why the function has no such asymptote.
(a)
\(\displaystyle f(x) = \frac{x^3 - 6x^2 + 5x}{x^2-1}\)
(b)
\(\displaystyle g(x) = \frac{11(x^2 + 1)(x-7)}{23(x-1)(x^2+4)}\)
(c)
\(\displaystyle h(x) = \frac{x^2 - 8x + 12}{x^2 - 3x - 18}\)
(d)
\(\displaystyle q(x) = \frac{(x-2)(x^2-9)}{(x-3)(x^2 + 4)}\)
(e)
\(\displaystyle r(x) = \frac{19(x-2) (x-3)^2 (x+1)}{17(x+1)(x-4)^2(x-5)}\)
(f)
\(\displaystyle s(x) = \frac{1}{x^2 + 1}\)
