Great question! At first glance, it might not seem intuitive that $g(n) = n^2$ could serve as an upper bound for $f(n) = 3n^2$, because $f(n)$ grows faster. However, this comes down to how we define "upper bounds" in the context of asymptotic analysis and Big-O notation.

Big-O Notation and Upper Bounds:

When we say that one function "upper bounds" another in Big-O notation, we're not comparing them on a point-by-point basis, but rather focusing on their behavior as $n$ grows very large. Specifically, we're interested in finding a function that grows at least as fast as $f(n)$ (or faster) in the long run, up to constant factors.

In this case:

• $f(n) = 3n^2$ and
• $g(n) = n^2$

The key idea is that Big-O notation ignores constant coefficients. So, $f(n) = 3n^2$ can be expressed as $O(n^2)$, because for large $n$, the factor of 3 becomes irrelevant when determining the growth rate.

Thus, $g(n) = n^2$ does indeed provide an upper bound for $f(n) = 3n^2$, because in terms of growth rate, they both grow quadratically, and we don’t care about the constant multiplier in asymptotic analysis.

Why Choose $g(n)$?

• Asymptotic Growth: Both $f(n)$ and $g(n)$ grow at the same rate, $O(n^2)$, which makes $g(n)$ a valid upper bound.
• Big-O Ignores Constants: The factor of 3 is irrelevant in Big-O notation, so $O(n^2)$ still captures the behavior of $f(n)$.
• Simplicity: We generally prefer to choose simpler upper bounds when possible. Since $g(n) = n^2$ is simpler than $3n^2$, it's a more convenient and valid choice.

Summary:

Choosing $g(n) = n^2$ to upper bound $f(n) = 3n^2$ makes sense within the framework of Big-O notation, where constant factors are ignored, and only the dominant term for large $n$ matters. Both functions grow at the same quadratic rate.

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Would you like more details on asymptotic analysis or Big-O notation? Here are 5 related questions to explore further:

1. How does Big-O notation differ from Big-Theta and Big-Omega?
2. What does it mean for a function to "dominate" another in terms of growth rate?
3. Can two functions have the same Big-O but behave differently for small $n$?
4. Why is constant time complexity written as $O(1)$?
5. How do you determine the Big-O of more complex functions, like polynomials or logarithms?

Tip: When using Big-O notation, always focus on the dominant term and ignore constants—they don't affect the asymptotic behavior!