Austin is using graphs to compare the growth rates of g(x)=1.3x and f(x)=1.3x . Which statement correctly describes how Austin should do this and what he will observe? (1 point)

Question

Responses

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

Austin should compare the steepness of the curves. The growth rate of g(x)=1.3x will quickly surpass the growth rate of f(x)=1.3x.
Austin should compare the steepness of the curves. The growth rate of g left parenthesis x right parenthesis equals 1.3 x will quickly surpass the growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline .

Austin should find where one curve is above the other curve on the graph. The growth rate of f(x)=1.3x is only greater than the growth rate of g(x)=1.3x 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 f(x)=1.3x will quickly surpass the growth rate of g(x)=1.3x.
Austin should compare the steepness of the curves. The growth rate of f left parenthesis x right parenthesis equals 1.3 superscript x baseline will quickly surpass the growth rate of g left parenthesis x right parenthesis equals 1.3 x .

Answer 1

It seems like there is a misunderstanding in the functions provided. Both functions, \( g(x) = 1.3^x \) and \( f(x) = 1.3^x \), are in fact the same. Thus, they will be identical across their entire domain, meaning their growth rates will be the same.

The correct statement would emphasize that since both functions are the same, they cannot be compared in terms of growth rates because they will overlap perfectly everywhere.

However, if the intention was to compare a linear function and an exponential function (for example, \( g(x) = 1.3^x \) vs. \( f(x) = 1.3x \)), then the correct approach would include noting that:

• The exponential function \( g(x) = 1.3^x \) will eventually outpace the linear function \( f(x) = 1.3x \) as \( x \) increases.

Given the provided options, none are accurate regarding the same functions. If you meant to compare an exponential function with a linear function, then the correct response would be something like:

“Austin should compare the steepness of the curves. The growth rate of the exponential function \( g(x) = 1.3^x \) will quickly surpass the growth rate of the linear function \( f(x) = 1.3x \).”

Please clarify if you intended different functions for comparison!