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4. Determine whether x3 is O(g(x)) for each of these functions g(x).a) g(x) = x2 b) g(x) = x3 c) g(x) = x2 + x3d) g(x) = x2 + x4 e) g(x) = 3x f ) g(x) = x3/2

Question

4. Determine whether x³ is O(g(x)) for each of these functions g(x).

a) g(x) = x²
b) g(x) = x³
c) g(x) = x² + x³
d) g(x) = x² + x⁴
e) g(x) = 3x
f) g(x) = x³/₂

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Solution

a) No, x^3 is not O(x^2) because x^3 grows faster than x^2 for large x.

b) Yes, x^3 is O(x^3) because they grow at the same rate.

c) Yes, x^3 is O(x^2 + x^3) because x^3 is the dominant term and determines the growth rate.

d) No, x^3 is not O(x^2 + x^4) because x^4 grows faster than x^3 for large x.

e) No, x^3 is not O(3x) because x^3 grows faster than 3x for large x.

f) No, x^3 is not O(x^3/2) because x^3 grows faster than x^3/2 for large x.

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