Marshall is comparing the growth rates of f(x) = 6 • 1.5* and g (x) = 5x + 10 using this

Question

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 • 1.52
°. Where is his mistake?
(1 point)
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 • 1.5* 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 right on the graph, he would observe that the growth rate of f(x) = 6 - 1.5* eventually
surpasses the growth rate of g (x) = 5x + 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 - 1.5* when x is greater than approximately -1.3.
Marshall has it backward. The growth rate of f(x) = 6 - 1.5* is always greater than the growth
rate of g (x) = 5x + 10.

Answer 1

Marshall's mistake lies in the region of the graph he is considering. The correct option 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 • 1.5^x eventually surpasses the growth rate of g(x) = 5x + 10 for large values of x.

This is because the function \(f(x) = 6 \cdot 1.5^x\) is an exponential function and will grow faster than the linear function \(g(x) = 5x + 10\) as \(x\) increases. Although \(g(x)\) may seem greater than \(f(x)\) for small values of \(x\), eventually \(f(x)\) will exceed \(g(x)\) for sufficiently large \(x\).