Question

Use the image to answer the question.

A coordinate plane shows an increasing solid curve and a dotted straight line plotted on it, corresponding to two functions.
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The x-axis ranges from negative 1.6 to 1.6 in increments of 0.2 and the y-axis ranges from negative 4 to 26 in increments of 2. The curved line labeled f left parenthesis x right parenthesis equals 6 times 1.5 superscript x passes through left parenthesis negative 1.6 comma 3 right parenthesis, left parenthesis negative 1 comma 4 right parenthesis, left parenthesis 0 comma 6 right parenthesis, left parenthesis 0.6 comma 7.5 right parenthesis, left parenthesis 1 comma 9 right parenthesis, and left parenthesis 1.7 comma 12 right parenthesis. The straight line labeled g left parenthesis x right parenthesis equals 5 times x plus 10 passes through left parenthesis negative 1.6 comma 2 right parenthesis, left parenthesis 0 comma 10 right parenthesis, and left parenthesis 1.6 comma 18 right parenthesis. Both the curve and the straight line have arrows at both ends. The line and the curve intersect at left parenthesis negative 1.3 comma 3.7 right parenthesis. All coordinate values are approximate, and the coordinates are unlabeled.


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 .

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 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 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 is looking at too small a portion of the domain. For large x the exponential's instantaneous growth rate surpasses the line's constant slope.

Reason: f'(x)=6(1.5)^x ln(1.5) and g'(x)=5. Solve 6(1.5)^x ln(1.5)>5 to get x ≈ 1.78. For x greater than about 1.78 the growth rate of f exceeds 5. So option 1 is correct.