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 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 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 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.

Answer 1

To correctly analyze the growth rates of the functions \( f(x) = 1.3^x \) and \( g(x) = 1.3x \), it is essential to understand the nature of exponential growth versus linear growth.

The best response from the options provided would be:

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

This statement accurately reflects that exponential functions grow faster than linear functions as \( x \) increases. Initially, \( g(x) = 1.3x \) may be greater for small values of \( x \), but eventually \( f(x) = 1.3^x \) will outpace and surpass it, exhibiting a much steeper growth rate as \( x \) continues to increase.