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.
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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.
