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Question

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.

Marshall is comparing the growth rates of f(x)=6⋅1.5x
and g(x)=5x+10 using this graph. Based on the graph, he concludes that the growth rate of g(x)=5x+10 is always greater than the growth rate of f(x)=6⋅1.5x

. Where is his mistake?
(1 point)
Responses

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f(x)=6⋅1.5x

eventually surpasses the growth rate of g(x)=5x+10 for large values of x
.
Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline eventually surpasses the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 for large values of x .
Marshall has it backward. The growth rate of f(x)=6⋅1.5x
is always greater than the growth rate of g(x)=5x+10
.
Marshall has it backward. The growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline is always greater than the growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 .
The growth rate of g(x)=5x+10
is only greater than the growth rate of f(x)=6⋅1.5x when x is greater than approximately −1.3
.
The growth rate of g left parenthesis x right parenthesis equals 5 x plus 10 is only greater than the growth rate of f left parenthesis x right parenthesis equals 6 times 1.5 superscript x baseline when x is greater than approximately negative 1.3 .
Marshall is only considering a small portion of the domains of the functions. If he went farther to the left on the graph, he would observe that the growth rate of f(x)=6⋅1.5x
eventually surpasses the growth rate of g(x)=5x+10 for negative values of x.

Answer 1

Marshall's mistake lies in his assessment that the growth rate of \( g(x) = 5x + 10 \) is always greater than that of \( f(x) = 6 \cdot 1.5^x \). The correct response to indicate his error is:

Marshall is only considering a small portion of the domains of the functions. If he went farther to the right on the graph, he would observe that the growth rate of \( f(x)=6 \cdot 1.5^x \) eventually surpasses the growth rate of \( g(x)=5x+10 \) for large values of x.

This option accurately describes the behavior of the two functions as \( x \) increases, where exponential growth (\( f(x) \)) will eventually outpace linear growth (\( g(x) \)).