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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)

Responses

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 .

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 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 g(x)=1.3x will quickly surpass the growth rate of f(x)=1.3x.

1 answer

To accurately compare the growth rates of the functions \( g(x) = 1.3^x \) and \( f(x) = 1.3x \), Austin should indeed analyze the behavior of the functions as \( x \) increases.

  1. Understanding the Functions: The function \( f(x) = 1.3^x \) is an exponential function, whereas \( g(x) = 1.3x \) is a linear function. Exponential functions grow faster than linear functions as \( x \) becomes large.

  2. Observations about Growth Rate: As \( x \) increases, \( 1.3^x \) will eventually exceed \( 1.3x \) and will grow significantly faster.

Considering these points, the appropriate response is:

"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 observation highlights the exponential growth outpacing linear growth, which is key to understanding their behaviors.