I'm trying to figure out what is growth of a function and growth rate of a function means and how should I note it.
As I understand growth rate of f(n) is dominant term of f(n) when n tends to infinity. So if we have for example:$$f(n) = 4n^2 + 5 + 7n^3$$
The growth rate of f(n) will be $$ growthRate=7n^3$$ where growthRate as I understand should be noted in some formal way.
And I have no idea what growth of a function means.
1 Answer
$\begingroup$In simple terms, growth rate of a function $f(x)$ means how fast the value of $f(x)$ increasing or decreasing as the value of $x$ increases. For example, if $f(x) = x$, for every unit increase in $x$, the function increases by one unit but if if $f(x) = 10x$, then for every unit increase in $x$, the function increases by 10 units.
In your example $$f(n) = 4n^2 + 5 + 7n^3$$observe that for $|n| > 1$ the magnitude of $n^3$ is always greater than that of $n^2$. Hence as $n$ increases, the corresponding increase in $n^3$ will be greater than that of $n^2$. Hence we say that growth rate of this function is of the order of $n^3$ and the exact growth rate is $7n^3$. Note that when we use the term order, we are only interested in the largest power on $n$ which in this case is 3 and we are not interested in the coefficient.
Explanation in layman's terms: If your salary is \$8,523 per month and someone asks you what your salary is you can either say the exact amount \$8529 or you can \$8000 because compared to \$8000, the remaining amount \$529 is relatively small and you can ignore it in you day to day conversation and still give a fairly accurate answer just by saying \$8000.
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