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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 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 .

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 g(x)=1.3x
is greater than the growth rate of f(x)=1.3x
between the intersection points of the curve.

1 answer

To analyze the growth rates of the functions \( g(x) = 1.3x \) and \( f(x) = 1.3^x \), Austin should compare the steepness of the curves. The important observation here is that \( f(x) = 1.3^x \) is an exponential function, which grows much faster than the linear function \( g(x) = 1.3x \) for larger values of \( x \).

The correct statement that describes how Austin should compare the growth rates 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 \( f(x) = 1.3^x \) increases exponentially, while \( g(x) = 1.3x \) increases linearly. As \( x \) increases, \( f(x) \) will become significantly steeper than \( g(x) \).