Find the big O of f(x) = 3x² + log(x)7 Prove that f(x) = x³ ‡ O(xlog(x²))
Ans 1:
The big O notation is used to describe the upper bound of the growth rate of a function. It provides an estimation of how the running time of an algorithm or function will increase as the input size grows. The goal is to determine the dominant term in the function's expression, which will determine its behavior as the input size becomes very large.
In this case, the function f(x) = 3x^2 + log(x)^7 can be analyzed as follows:
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The term 3x^2 grows very quickly as x increases. For example, when x = 10, the value of 3x^2 is 300, and when x = 100, the value is 30,000. This growth rate is proportional to x^2, which is a polynomial function with degree 2. Therefore, the term 3x^2 is a dominant term in the function and its big O is O(x^2).
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The term log(x)^7 grows much more slowly than x^2. For example, when x = 10, the value of log(x)^7 is about 6.6, and when x = 100, the value is about 9.3. The logarithmic function grows much more slowly than the polynomial function, so its big O is O(log(x)).
Based on this analysis, the big O of the function f(x) = 3x^2 + log(x)^7 is O(x^2). This means that the running time of the function will increase proportionally to x^2 as the input size grows. The logarithmic term is not dominant and can be ignored in the big O analysis. So its big O becomes O(log(x)).
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