Suppose that fx gx and hx are functions such that fx is bigT

Suppose that f(x), g(x), and h(x) are functions such that f(x) is big-Theta (g(x)) and g(x) is big-Theta (h(x)). Show that f(x) is big-Theta(h(x)). (Discrete Mathematics)

44. Suppose that f(x), g(x), and h(x) are functions such that f(x) is erg(x)) and g(x) is (h(x)). Show that f(x) is (h(x))

Solution

Since f(x) is (g(x)), there exist constants A, B > 0 such that
A * g(x) < f(x) < B * g(x) for sufficiently large x.

Similarly, since g(x) is (h(x)), there exist constants C, D > 0 such that
C * h(x) < g(x) < D * h(x) for sufficiently large x.

Hence, for sufficiently large x, we have
f(x) < B g(x) < B * (D * h(x)) = (BD) h(x), and
f(x) > A g(x) > A * (C * h(x)) = (AC) h(x).

Hence, we have constants E, F > 0 (where E = AC and F = BD) such that
E * h(x) < f(x) < F * h(x) for sufficiently large x.

Therefore, f(x) is (h(x)).