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?

Question

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Responses

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.
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 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 negative 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 .

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 .
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Answer 1

Marshall's mistake lies in his conclusion about the growth rates of the functions as x increases. The correct response 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⋅1.5x eventually surpasses the growth rate of g(x)=5x+10 for large values of x.

This statement highlights that while g(x) may initially grow faster than f(x) for certain values of x, the exponential growth rate of f(x) will eventually outpace the linear growth rate of g(x) as x increases.