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

Question

. 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 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 .
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.

Answer 1

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

The correct observation is that the exponential function \( g(x) = 1.3^x \) grows much faster than the linear function \( f(x) = 1.3x \) for large values of \( x \). Initially, for smaller values of \( x \), \( f(x) \) may be greater than \( g(x) \), but as \( x \) increases, \( g(x) \) will eventually surpass and then continue to grow faster than \( f(x) \).

Therefore, the correct choice among the provided responses is:

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

This statement correctly describes how to compare the two functions and what observation he should expect regarding their growth rates.