Show that if f and g are real-valued functions such that f(x) is O(g(x)), then for every positive integer n, fn(x) is O(gn(x)). [Note that fn(x) = f(x)n.]

Step1:

In this problem, we have to prove that for every positive integer n, fn(x) is O(gn(x)) [Note that fn(x) = f(x)n.]

Step 2:

Let us consider if f and g are real-valued functions such that f(x) is O(g(x))

We can say that f(x) is O(g(x)) if there are constant C and k such that |f(x)||g(x)| whenever x>k

Similarly for this case f(x) is O(g(x)) there are constant C and 1

such that |f(x)|