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

Question

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 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 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 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 .
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Answer 1

The correct 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 is because \( g(x) = 1.3x \) represents a linear function while \( f(x) = 1.3^x \) represents an exponential function. Exponential functions grow faster than linear functions as \( x \) increases, so Austin will observe that \( f(x) \) surpasses \( g(x) \) beyond a certain point.