Question

Rank the following functions by increasing order of growth:

1. log( n! )
2. 10000n²
3. log(n³)
4. 2ⁿ
5. n²log(n)

Answer 1

The functions ranked by increasing order of growth are log(n!), n²log(n), log(n³), 10000n², and 2ⁿ.

Step-by-step explanation:

1. Rank the functions by increasing order of growth:
2. log( n! ) - This function has a logarithmic growth rate, which means the growth rate of the function decreases as n increases. Therefore, it has the slowest growth rate among the given functions.
3. n²log(n) - This function has a quadratic growth rate, which means the growth rate of the function increases as n increases. Therefore, it has a faster growth rate than log(n!).
4. log(n³) - This function has a growth rate between logarithmic and quadratic. It increases slower than n²log(n) but faster than log(n!).
5. 10000n² - This function has a quadratic growth rate, which means the growth rate of the function increases at a faster pace compared to log(n³).
6. 2ⁿ - This function has an exponential growth rate, which means the growth rate of the function increases significantly as n increases. Therefore, it has the fastest growth rate among the given functions.