Final answer:
To arrange the given functions in order of increasing rate of growth, we analyze the growth rates of each function and compare them. We find that the correct order is n², 2ⁿ, 4ⁿ, nⁿ, n!, 2²ⁿ. The functions nⁿ and n! have the same rate of growth.
Step-by-step explanation:
To arrange the functions in order of increasing rate of growth, we need to determine the growth rates of each function. Let's analyze each function:
- n²: This function has a quadratic growth rate, which means that as the input value increases, the output value increases quadratically (the rate of increase is proportional to the square of the input).
- 2ⁿ: This function has an exponential growth rate, where the output value grows exponentially as the input increases. The rate of increase is much faster than quadratic growth.
- n!: This function represents factorial growth, where the output value increases rapidly as the input value increases. The rate of increase is even faster than exponential growth.
- 4ⁿ: This function also has an exponential growth rate, but with a different base. The rate of increase is faster than quadratic growth but slower than factorial growth.
- nⁿ: This function has an even faster growth rate than factorial growth. The output value grows rapidly as the input increases.
- 2²ⁿ: This function represents an exponential function with a larger base than the previous functions. The rate of increase is the fastest among all the functions.
Based on these growth rates, we can arrange the functions in increasing order: