Question

1a. If

$$f(x) = \frac{2x-8}{x^2 -2x - 3} \qquad\text{ and }\qquad g(x) = \frac{3x+9}{2x-4}$$find the sum of the values of $x$ where the vertical asymptotes of $f(g(x))$ are located.

1b. What is the horizontal asymptote as $x$ approaches negative infinity of $f(g(x))$?

(Don't forget that an asymptote is a LINE, and not a NUMBER!)

Answer 1

The function f(g(x)) eventually simplifies to
f(g(x) = (4/3)(x-5)(x-2)/(x² - 6x - 7)

a) The sum of the x-coordinates of the vertical asymptotes is the opposite of the x-coefficient in the denominator: 6.

b) Since the numerator and denominator are of the same degree, the horizontal asymptote is the overall multiplier: y = 4/3.


_____
Consider a quadratic with roots at "a" and "b". Then the factored form is
(x-a)(x-b)
and the expanded form is
(x² - (a+b)x + ab)
That is, the sum of the roots is the opposite of the x-coefficient. For this problem, that sum is the sum that is requested in part A.