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 only 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

The correct statement that describes how Liam should make the comparison and what he will observe 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.

Here's why this is the correct choice:

• The function \( h(x) = 1.2x \) is a linear function, while \( j(x) = 1.2^x \) is an exponential function.
• Exponential functions grow significantly faster than linear functions as \( x \) increases. Although \( h(x) \) may have higher values for small \( x \), eventually, \( j(x) \) will exceed \( h(x) \) and continue to grow at a much faster rate.
• Comparing the rates of change reveals that the derivative of \( j(x) \) (which is related to its exponential growth) increases faster than that of \( h(x) \) (which is constant).

Therefore, option C correctly summarizes Liam's method and the observation he will make.