Austin is using graphs to compare the growth rates of g(x)=1.3x

Question

𝑔
(
𝑥
)
=
1.3
𝑥
and f(x)=1.3x
𝑓
(
𝑥
)
=
1.3
𝑥
. Which statement correctly describes how Austin should do this and what he will observe? (1 point)
Responses

Austin should find where one curve is above the other curve on the graph. The growth rate of f(x)=1.3x
𝑓
(
𝑥
)
=
1.3
𝑥
is only greater than the growth rate of g(x)=1.3x
𝑔
(
𝑥
)
=
1.3
𝑥
to the right of their right-most intersection point.
Austin should find where one curve is above the other curve on the graph. The growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline is only greater than the growth rate of g left parenthesis x right parenthesis equals 1.3 x to the right of their right-most intersection point.

Austin should compare the steepness of the curves. The growth rate of g(x)=1.3x
𝑔
(
𝑥
)
=
1.3
𝑥
will quickly surpass the growth rate of f(x)=1.3x
𝑓
(
𝑥

Answer 1

To compare the growth rates of the functions \( g(x) = 1.3^x \) and \( f(x) = 1.3^x \), we first need to clarify: both functions are identical, and therefore their growth rates are the same for all values of \( x \).

However, assuming you meant to compare different functions (for example, \( g(x) = 1.3^x \) and \( f(x) = x \) or another function), here’s a corrected observation if those were the intended functions:

Corrected Explanation: Austin should find where one curve is above the other curve on the graph. If comparing \( g(x) = 1.3^x \) (an exponential function) and \( f(x) = x \) (a linear function), he will observe that \( g(x) \) will eventually surpass \( f(x) \) as \( x \) increases, indicating that the growth rate of the exponential function \( g(x) \) is greater than the linear growth of \( f(x) \).

If the functions are indeed both \( g(x) = 1.3^x \) and \( f(x) = 1.3^x \), then they are equal everywhere, so you would clarify that there is no point where one grows faster than the other, as they are the same function.

Make sure the functions are clearly indicated if they differ for accurate growth rate comparisons!

Answer 2

answer

Answer 3

Based on the context of your question, here is the correct response:

Austin should find where one curve is above the other curve on the graph. The growth rate of \( f(x) = 1.3^x \) is only greater than the growth rate of \( g(x) = 1.3^x \) to the right of their right-most intersection point.

However, if we consider that both functions are actually the same (i.e., \( g(x) = f(x) = 1.3^x \)), then neither function's growth rate is greater than the other because they are identical.

Please clarify the functions you're comparing if they're not both \( 1.3^x \).