This segment meticulously analyzes two piecewise-defined functions, g1 and g2, determining their relative magnitudes for different ranges of n. It emphasizes the importance of considering the behavior of functions across various intervals and the concept of asymptotic dominance for large values of n. This text compares function growth rates using logarithms to determine which function is asymptotically larger. It emphasizes that for large n, constant coefficients are often negligible. Examples illustrate comparing n log n vs. 2<sup>√n</sup>, and piecewise functions like n³ for n<10000 and n² thereafter. The analysis highlights the importance of considering asymptotic behavior for large n, not just small values. This segment demonstrates a step-by-step comparison of two logarithmic functions (applying log multiple times to simplify and compare) to determine which function is greater, highlighting the importance of careful analysis and simplification techniques in asymptotic notation.