Let f(n) = (n + a)b and g(n) = nb, for any real constants a and...

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Let f(n) = (n + a)b and g(n) = nb, for any real constants a and...

Let f(n) = (n + a)^b and g(n) = n^b, for any real constants a and b, where b > 0.

Using definition of O and g(n), establish the upper bound of f(n). Find the values of positive constant c and nonnegative integer n0 ,

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Answer 1

Answer #1

f(n) = O(g(n)) means there are positive constants c and n0, such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0

(n + a)b = O(nb)
=>  nb + ab = O(nb)
=>  nb + ab <= c(nb)
Let's assume c = 2
=>  nb + ab <= c(nb)
=>  nb + ab <= 2(nb)
=>  ab <= nb
=>  a <= n

it's true for all n >= a
so, (n + a)b = O(nb) given c = 2 and n0 = a

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Answer 2

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