Liam is using sequences to compare the growth rates of \( h(x) = 1.2x \) and \( j(x) = 1.2^x \). Which statement correctly describes how Liam should do this and what he will observe?

Question

A. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of \( j(x) = 1.2^x \) is only greater than the growth rate of \( h(x) = 1.2x \) when its terms are greater.

B. Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of \( h(x) = 1.2x \) is greater than the growth rate of \( j(x) = 1.2^x \) when its terms are greater.

C. Liam should compare the rates of change of the terms in both sequences. The growth rate of \( j(x) = 1.2^x \) will quickly surpass the growth rate of \( h(x) = 1.2x \).

D. Liam should compare the rates of change of the terms in both sequences. The growth rate of \( h(x) = 1.2x \) will quickly surpass the growth rate of \( j(x) = 1.2^x \)

Answer 1

To compare the growth rates of the two functions \( h(x) = 1.2x \) (a linear function) and \( j(x) = 1.2^x \) (an exponential function), Liam should indeed focus on the nature of their growth.

Exponential functions grow at a rate that increases as \( x \) increases, while linear functions grow at a constant rate. Specifically, \( j(x) = 1.2^x \) will eventually grow much faster than \( h(x) = 1.2x \) as \( x \) becomes large enough.

Therefore, the appropriate response is:

C. Liam should compare the rates of change of the terms in both sequences. The growth rate of \( j(x) = 1.2^x \) will quickly surpass the growth rate of \( h(x) = 1.2x \).