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 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 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
.

Answer 1

To compare the growth rates of the functions \( f(x) = 1.3^x \) and \( g(x) = 1.3x \), Austin should be aware that:

• \( f(x) = 1.3^x \) is an exponential function, which grows faster than any linear function as \( x \) increases.
• \( g(x) = 1.3x \) is a linear function.

Thus, the correct statement 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 choice accurately describes the behavior of the functions as \( x \) increases, confirming that while \( g(x) \) may be larger for small values of \( x \), \( f(x) \) will eventually outpace \( g(x) \) as \( x \) becomes larger.