Determine whether x^3 is O(g(x)) for each of these functions g(x).

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`Determine whether x^3 is O(g(x)) for each of these functions g(x).`

d) g(x) = x^{2} + x^{4}

e) g(x) = 3^{x}

f) g(x) = x^{3}/2

d) Yes

e) Yes

f) No

Are my answers correct? I took the problem to mean that g(x) < c * x^{3}.

Re: Determine whether x^3 is O(g(x)) for each of these functions g(x).

Hey lamentofking.

I think you have it the other way around. 3^x is an exponential function with infinite numbers of positive powered terms and x^4 has a term higher than x^3. f should be correct (i.e. yes).

Re: Determine whether x^3 is O(g(x)) for each of these functions g(x).

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Originally Posted by

**lamentofking** Determine whether x^3 is O(g(x)) for each of these functions g(x).

I took the problem to mean that g(x) < c * x^{3}.

No, it means that eventually x^{3} < c * g(x).

Re: Determine whether x^3 is O(g(x)) for each of these functions g(x).

Quote:

Originally Posted by

**emakarov** No, it means that eventually x^{3} < c * g(x).

So then in terms of the question d), e), and f) are all yes? I thought f) would be no because if x3 < c * g(x) and (x^3)/2 means (1/2)x^3 with c=1/2 then x^3 would be greater than (>) c* g(x).

Re: Determine whether x^3 is O(g(x)) for each of these functions g(x).

Quote:

Originally Posted by

**lamentofking** So then in terms of the question d), e), and f) are all yes?

Yes.

Quote:

Originally Posted by

**lamentofking** I thought f) would be no because if x3 < c * g(x) and (x^3)/2 means (1/2)x^3 with c=1/2 then x^3 would be greater than (>) c* g(x).

Could you please re-formulate this clearly? "If x^3 < c * g(x)" eventually, then x^3 = O(g(x)), nothing more to prove. Are you sure you are *assuming* x^3 < c * g(x)? Next, by c * g(x) do you mean (1/4)x^3 since c = 1/2 and g(x) = (1/2)x^3? That's possible, but why not consider c = 1? Your quote above is not very clear.