Liam is using sequences to compare the growth rates of h(x)=1.2x

Question

and j(x)=1.2x
. Which statement correctly describes how Liam should do this and what he will observe?(1 point)
Responses

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)=1.2x
.
Liam should compare the rates of change of the terms in both sequences. The growth rate of h left parenthesis x right parenthesis equals 1.2 x will quickly surpass the growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline .

Liam should compare the rates of change of the terms in both sequences. The growth rate of j(x)=1.2x
will quickly surpass the growth rate of h(x)=1.2x
.
Liam should compare the rates of change of the terms in both sequences. The growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline will quickly surpass the growth rate of h left parenthesis x right parenthesis equals 1.2 x .

Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j(x)=1.2x
is only greater than the growth rate of h(x)=1.2x
when its terms are greater.
Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline is only greater than the growth rate of h left parenthesis x right parenthesis equals 1.2 x when its terms are greater.

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)=1.2x
when its terms are greater.
Liam should look at where one sequence has terms greater than the terms in the other sequence. The growth rate of h left parenthesis x right parenthesis equals 1.2 x is greater than the growth rate of j left parenthesis x right parenthesis equals 1.2 superscript x baseline when its terms are greater.

Answer 1

To correctly describe how Liam should compare the growth rates of the functions \( h(x) = 1.2^x \) and \( j(x) = 1.2x \), the key point is that \( h(x) \) is an exponential function, while \( j(x) \) is a linear function.

As \( x \) increases, exponential functions grow much faster than linear functions. Therefore, the correct statement would be:

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 \( h(x) = 1.2x \).

This highlights the fact that as you move to larger values of \( x \), the growth of \( h(x) \) (the exponential function) will quickly become much larger than that of \( j(x) \) (the linear function). Therefore, the best option from the responses you've provided would be:

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 \( h(x) = 1.2x \).