The rational function f(x) = p(x)
q(x)
, where p and q are quadratic functions, has the following proper-
ties:
(a) zeros at 3 and -1;
(b) vertical asymptotes x = 1 and x = 4;
(c) a horizontal asymptote y = 4.
Determine f(2).

Question

The rational function f(x) = p(x)
q(x)
, where p and q are quadratic functions, has the following proper-
ties:
(a) zeros at 3 and -1;
(b) vertical asymptotes x = 1 and x = 4;
(c) a horizontal asymptote y = 4.
Determine f(2).

anonymous · Answer

p(x)/q(x)

blockcolder · Answer

Let's incorporate all the clues.
If f has 0's at 3 and -1, this means that x-3 and x+1 are factors of p(x). Therefore, p(x)=A(x+1)(x-3).
If f has VA's at 1 and 4, this means that x-1 and x-4 are factors of q(x). Therefore, q(x)=B(x-1)(x-4).
An HA at y=4 means that the ratio A/B is 4.
From all of these clues, we gather that
$$f(x)=\frac{4(x-3)(x+1)}{(x-1)(x-4)}$$
and thus
$$f(2)=\frac{4(-1)(3)}{(1)(-2)}=6$$